---
abstract: 'We introduce a method to estimate the size of the domain of definition of the solutions of a meromorphic vector field on a neighborhood of its pole divisor. The corresponding techniques are, in a certain sense, quantitative versions of some well-known phenomena related to the presence of metrics with positive curvature. Several applications of these ideas are provided including a type of “confinement theorem” for solutions of complete polynomial vector fields on ${{\mathbb C}}^n$ and obstructions for certain (germs of) vector fields to be realized by a global holomorphic vector field on a compact Kähler manifold. As a complement a new approach to certain classical equations is proposed and detailed in the case of Halphen equations.'
author:
- 'Julio C. Rebelo & Helena Reis'
title: Uniformizing complex ODEs and applications
---

Introduction
============

The object of this paper is a method to investigate the domain of definition of solutions for holomorphic (meromorphic) vector fields. This method is quite general and applicable to arbitrarily high dimensions whereas it provides new results already in dimension $3$. As a consequence this work is essentially constituted by two parts, the first one corresponding to the general setting and central results which will then be exploited in the second part to yield the mentioned applications. To greater or lesser extent, these applications arise from following the solution of a complex (polynomial or rational) vector field over “special real paths going off to infinity”.

This Introduction is aimed only at stating the main applications considered in this work. These results were chosen not with a purpose of being the sharpest possible but rather of indicating ways of exploiting our method which is considered to be the main contribution of this work. In Section 2 we shall provide a more detailed discussion explaining our point of view and underlining the common structures lying behind the theorems below. It is also to be noted that our applications concern very special types of vector fields (or of differential equations) such as complete vector fields and Halphen equations. Nonetheless the setting is also well-adapted to investigating differential equations having meromorphic solutions including several classical equations appearing in Mathematics and Physics. For example equations concerned with the works of Painlevé and Chazy are among those having a very immediate connection with our work. Whereas these possibilities are not developed here, they seem to provide a clear indication that further applications of our techniques will be found in a near future.

Let us begin by considering a complete polynomial vector field $X$ on ${{\mathbb C}}^n$ of degree at least $2$. This means that its complex solutions are defined for all $T \in {{\mathbb C}}$. In the special case in which $X$ is also completely integrable, its orbits can be compactified into rational curves by adding to them some “singular points of $X$ at infinity”. This fact can be interpreted as a type of confinement phenomenon for the corresponding solutions. Our first results can be regarded as a weaker, whereas sharp, type of confinement phenomenon holding for every solution of a complete polynomial vector field. To state it, we proceed as follows. Since $X$ is polynomial it defines a singular holomorphic foliation ${\mathcal{D}}$ on ${{\mathbb C}}P(n) = {{\mathbb C}}^n \cup \Delta_{\infty}$ viewed as a compactification of ${{\mathbb C}}^n$. Consider a leaf $L$ of ${\mathcal{D}}$ (details on the definition of “leaf” in the singular context can be found in Section 3). On $L$ two (singular) oriented real one-dimensional foliations ${\mathcal{H}}, \, {\mathcal{H}}^{\perp}$ are going to be defined. They will depend on the leaf $L$ of ${\mathcal{D}}$ is a regular way as it will be apparent from their definitions (cf. Sections 3 and 6). More importantly, ${\mathcal{H}}, \, {\mathcal{H}}^{\perp}$ are mutually orthogonal with respect to the conformal structure of $L$, in fact, they agree respectively with the real foliation and the purely imaginary foliation induced on $L$ by a certain Abelian form. Since $L$ is endowed with a conformal structure, it makes sense to define also foliations ${\mathcal{H}}^{\theta}$ whose (oriented) trajectories makes an angle $\theta$ with the oriented trajectories of ${\mathcal{H}}$ ($\theta \in [-\pi /2, \pi /2]$). The trajectories of these foliations define the “directions of confinement” for $L$ as it is seen from Theorem A below. In the sequel $\Phi : {{\mathbb C}}\times {{\mathbb C}}^n \rightarrow {{\mathbb C}}^n$ stands for the holomorphic flow generated by $X$ whereas ${\rm Sing}\, ({\mathcal{D}})
\subset {{\mathbb C}}P(n)$ denotes the singular set of ${\mathcal{D}}$.

[**Theorem A**]{}. [*Suppose that $X$ is a complete polynomial vector field as above. Fix an arbitrarily small neighborhood $V$ of $({\rm Sing}\, ({\mathcal{D}}) \cap \Delta_{\infty}) \cup
{\rm Sing}\, (X)$ in ${{\mathbb C}}P(n)$ and suppose we are given a point $p \in {{\mathbb C}}^n$, $X (p) \neq 0$ and an angle $\theta \in
(-\pi/2 ,\pi/2)$. Denote by $L_p$ (resp. $l_p^{+,\theta}$) the leaf of ${{\mathcal F}}$ through $p$ $($resp. the semi-trajectory of ${\mathcal{H}}^{\theta}$ initiated at $p)$ and consider the lift $c:[0, \infty) \rightarrow {{\mathbb C}}$ of $l_p^{+,\theta}$ by $\Phi$, i.e. $t \in [0, \infty) \mapsto
\Phi (c(t), p)$ is a one-to-one parametrization of $l_p^{+,\theta}$ $(c(0)=0)$. Then there is a constant $C$ such that $${\rm meas} \, \left( \{ t \in [0, \infty) \; ; \;  \Phi (c(t), p) \not\in V \} \right) < C \, ,$$ where ${\rm meas}$ stands for the usual Lebesgue measure on ${{\mathbb R}}$.* ]{}

The preceding theorem states that the trajectory $l_p^{+,\theta}$ spend most of its “life” in the neighborhood $V$ and hence arbitrarily close to the singular points of ${\mathcal{D}}$. Furthermore the constant $C$ varies continuously with $\theta$. In particular if we consider a compact interval $[-\pi/2 + \delta ,
\pi /2 - \delta] \subset (-\pi/2 ,\pi/2)$ then $C$ can be chosen so that the above estimate holds for every $\theta \in [-\pi/2 + \delta , \pi /2 - \delta]$. The existence of $C$ uniform for $\theta \in [-\pi/2 + \delta , \pi /2 - \delta]$ allows us to generalize the statement to paths $c \subset L_p$ more general than the trajectories of ${\mathcal{H}}^{\theta}$, for example we may consider paths $c$ as before such that the angle made at the point $c(t)$ by the speed vector $c'(t)$ and the foliation ${\mathcal{H}}$ lies in $[-\pi/2 + \delta , \pi /2 - \delta]$ for all $t$. Details on these statements are left to the reader.

Confinement phenomena contrast with ergodicity so that it is natural to search for a variant of this theorem focusing on the “area” defined in ${{\mathbb C}}$ by those values of $T$ for which $\Phi (T,p) \in V$. This variant might be viewed, in particular, as a “super non-ergodic” phenomenon for complete vector field. To state it, let $B_r \subset {{\mathbb C}}$ denote the disc of radius $r$ about $0 \in {{\mathbb C}}$. A continuous path properly embedded $c: (-\infty, \infty) \rightarrow {{\mathbb C}}$ is a [*separating curve*]{} if it is of class $C^{\infty}$ with possible exception of a discrete set and it is either periodic or it satisfies the condition $\lim_{t \rightarrow -\infty} c(t) = \infty$, $\lim_{t \rightarrow \infty} c(t) = \infty$ (where the last conditions means that the curve eventually leaves every compact subset of ${{\mathbb C}}$. A separating curve divides ${{\mathbb C}}$ in at least two connected component and at least one of these components is unbounded. Then we have:

[**Theorem A’**]{}. [*Let $X$, $V$, $L_p$ and $p \in {{\mathbb C}}^n$ be as above. Consider the parametrization of $L_p$ by ${{\mathbb C}}$ (possibly as a covering map) which is given by $\Phi_p (T) = \Phi (T, p)$. Then there exists a separating curve $c: (-\infty, \infty) \rightarrow {{\mathbb C}}$, $\Phi_p(c(0)) =p$, and an unbounded component $\mathcal{U}^+$ of ${{\mathbb C}}\setminus c(t)$ such that the following holds: the set $\mathcal{T}_V \subset \mathcal{U}^+ \subset {{\mathbb C}}$ defined by $$\mathcal{T}_V = \{ T \in \mathcal{U}^+ \subset {{\mathbb C}}\; ; \; \Phi (T,p) \in V \} \,$$ satisfies $$\lim_{r \rightarrow \infty} \frac{ {\rm Meas}\, (\mathcal{T}_V \cap B_r)}
{{\rm Meas}\, (\mathcal{U}^+ \cap B_r)} =1 \, ,$$ where ${\rm Meas}$ stands for the usual Lebesgue measure of ${{\mathbb C}}$ ($\simeq {{\mathbb R}}^2$).*]{}

Unlike most standard averaging theorems, the statement above holds for every point $p \in {{\mathbb C}}^n$ and not only for almost all points. Besides it is easy to conclude from the proof given in Section 6.1 that for almost all points $p$ the corresponding separating curve is smooth. In fact, this separating curve is nothing but a geodesic (“straight line”) for a suitable flat structure on ${{\mathbb C}}$ which is naturally comparable with the standard one (for further details see Section 6.1). In particular it follows that ${\rm Meas}\, (\mathcal{U}^+ \cap B_r)$ is, in fact, comparable to the measure of the large discs $B_r$. Since discs of this nature are used in the construction of Ahlfors currents, the previous statement may look unnatural since the latter cannot charge singular points. Explanation for this difference is however easy since the construction of Ahlfors currents is based on the “global volume” and this may have little relation with the asymptotic behavior of an actual solution. To be more precise fix a diffeomorphism between ${{\mathbb C}}$ and a leaf $L$, for example a time-$t$ diffeomorphism $\Phi_t$ induced by the corresponding vector field. To construct Ahlfors current the ambient manifold is equipped with a Hermitian metric which is then pulled back by $\Phi_t$ to yield a metric $d_{{{\mathbb C}}}$ on ${{\mathbb C}}$. Ahlfors current is then constructed by choosing a suitable sequence of discs $B_{r_i}$ whose radii $r_i$ are measured with respect to $d_{{{\mathbb C}}}$ and satisfy $r_i \rightarrow \infty$. Clearly a “small” neighborhood $V$ of a singular point in $M$ has small diameter for the fixed Hermitian metric and so does a connected component $(L \cap V)_0$ of $L \cap V$. The diameter (resp. the “area”), of $\phi_t^{-1} ((L \cap V)_0)$ w.r.t. $d_{{{\mathbb C}}}$ is therefore small as well. Now we should note that $d_{{{\mathbb C}}}$ may differ markedly from the Euclidean metric on ${{\mathbb C}}$ thus it might happen, for example, that the euclidean area of $\phi_t^{-1} ((L \cap V)_0)$ is “large”. The proofs of the preceding theorems will make it clear that this phenomenon is precisely what happens in these cases. As a conclusion, whereas Ahlfors currents are among the most efficient tools for studying (singular) holomorphic foliations possessing leaves covered by ${{\mathbb C}}$, they might be less so when considering specific differential equations.

The statement of Theorems A and A’ indicate that the structure of the singularities of ${\mathcal{D}}$ lying in $\Delta_{\infty}$ must bear significant information on the global dynamics of corresponding vector fields. According to this principle, it is natural to wonder that complete vector fields whose associated foliations ${\mathcal{D}}$ have only “simple singularities” in $\Delta_{\infty}$ must be amenable to a detailed global analysis. For “simple” singularities we shall mean the following types of singular points $q \in \Delta_{\infty}$ for ${\mathcal{D}}$:

1.  Non-degenerate singularities: this means that ${\mathcal{D}}$ can locally be represented by a vector field having non-degenerate linear part at $q$ (i.e. the Jacobian matrix of $X$ at $q$ is invertible, equivalently, it possesses $n$ eigenvalues different from zero). Besides, since resonances may arise, we assume that $q$ is [*not of Poincaré-Dulac type*]{}, i.e. if all the eigenvalues of ${\mathcal{D}}$ at $q$ belong to ${{\mathbb R}}_+^{\ast}$ then ${\mathcal{D}}$ must be locally linearizable about $q$.

2.  Codimension $1$ saddle-nodes: these are singularities of ${\mathcal{D}}$ lying in $\Delta_{\infty}$ whose eigenvalue associated to the direction transverse to $\Delta_{\infty}$ is equal to [*zero*]{} whereas it has $n-1$ eigenvalues different from zero and corresponding to directions contained in $\Delta_{\infty}$. Again we require that the $(n-1)$-dimensional singularity induced on the plane $\Delta_{\infty}$ should not be a singularity of Poincaré-Dulac type.

Note that singular points of ${\mathcal{D}}$ as in item (1) above are necessarily isolated though this is no longer valid for Codimension $1$ saddle-nodes. Next we have:

[**Theorem B**]{}. [*Let $X$ be a complete polynomial vector field on ${{\mathbb C}}^n$ whose singular set has codimension at least $2$. Suppose that all singularities of ${\mathcal{D}}$ lying in $\Delta_{\infty}$ are as in items (1) or (2) above. Then the foliation ${\mathcal{D}}$ associated to $X$ defines a rational pencil on ${{\mathbb C}}P(n)$.*]{}

Theorem B will be proved in Section 6.2. The statement of this theorem may be compared to results of [@bruno] for complete polynomial vector fields on ${{\mathbb C}}^2$. It is to be noted that the results of [@bruno] chronologically preceded the classification obtained in [@marco3]. Also the recent paper [@guillotreb] contains a very general classification theorem for meromorphic vector fields admitting “maximal solutions” on complex surfaces and these include complete vector fields as in [@marco3]. All these questions are however wide open for $n \geq 3$ and our Theorem B appears as a contribution to them.

To have a better appreciation of the difficulties involved in these problems, let us consider the case of semi-complete vector fields (as considered for example in [@guillotreb]). Recall that a vector field is said to be semi-complete on a domain $U$ if its solution $\phi$ verifying $\phi (0) =p \in U$ is defined on a [*maximal domain*]{} of ${{\mathbb C}}$ for all $p \in U$. Here a domain $V \subseteq {{\mathbb C}}$ where the solution $\phi$ is defined is said to be [*maximal*]{} if for every point $\hat{T}$ in the boundary $\partial V$ of $V$ and every sequence $\{ T_i \} \subset V$ such that $T_i \rightarrow \hat{T}$, the sequence $\phi (T_i)$ leaves every compact set in $U$, cf. Section 2 for further details. Clearly a complete vector field is automatically semi-complete since we can take $V = {{\mathbb C}}$ so that $\partial V =\emptyset$. If we consider polynomial semi-complete vector fields on ${{\mathbb C}}^n$ then even the quadratic homogeneous case is already hard to understand once $n \geq 3$. In fact, A. Guillot has conducted detailed research about semi-complete quadratic homogeneous vector fields (with simple singularities at infinity) in [@guillotFourier], [@guillotIHES]. In [@guillotFourier] he introduced certain lattices (of coefficients) where all these vector fields are to be found whereas in [@guillotIHES] he studied the special case of Halphen’s vector fields and the related problem about actions of ${\rm PSL}\, (2, {{\mathbb C}})$ on compact $3$-manifolds. The beauty and depth of these results motivated us to try to apply our techniques to vector fields satisfying the conditions stated in [@guillotFourier]. To abridge notations, we shall refer to these vector fields by saying that they belong to Guillot lattice (the reader interested in the case $n = 3$ will find the document [@guillotThesis] particularly useful).

Note that a semi-complete vector field of ${{\mathbb C}}^3$ belonging to Guillot lattice may not be complete and its orbits (thought of as leaves of the associated foliation) may be hyperbolic Riemann surfaces. This is actually what occurs with Halphen vector fields except for a few special cases cf. [@guillotIHES], [@guillotThesis] or yet Section 7. In this Introduction by a Halphen vector field it is always meant a “hyperbolic” Halphen vector field. These vector fields are semi-complete and the maximal domains of definitions for their solutions may be either a bounded region of ${{\mathbb C}}$ or a (hyperbolic) unbounded region (for example the complement of a suitable disc).

More generally [@guillotFourier] contains a classification of semi-complete quadratic vector fields up to a finite number of vector fields. A full understanding of the dynamics of these possible “exceptional” cases seems to be a very hard problem. In any event, among vector fields belonging to Guillot lattice we shall also consider those satisfying the following condition: no singularity of the associated foliation lying in the hyperplane at infinity has all its eigenvalues lying in ${{\mathbb R}}_+$. They will be referred to as vector fields having no dicritical singularity at infinity. We can now state Theorem C.

[**Theorem C**]{}.

*Suppose that $X$ is a holomorphic vector field defined compact manifold $M$. Consider a singularity $p \in M$ of $X$ and denote by $X_k$ the first non-zero homogeneous component of the Taylor series of $X$ at $p$. Suppose that one of the following condition holds:*

$\bullet$ $X_k$ is a multiple of a vector field in Guillot lattice having no dicritical singularity at infinity.

$\bullet$ $X_k$ is a multiple of a hyperbolic Halphen vector field (in case $M$ has dimension $3$).

Then $M$ does not a carry a Kähler structure.

Note that Theorem C is somehow sharp in the sense that [@guillotIHES] contains examples of compact $3$-manifolds equipped with a global holomorphic vector field exhibiting the singularity of a hyperbolic Halphen vector field. Naturally the corresponding manifolds are not Kähler.

Theorem C will be proved in Section 7. The proof is indeed very short though based on the material developed in the preceding sections. The rest of the section will be taken up by a discussion of the main dynamical issues associated to Halphen vector fields. The results presented there are definitely not new as they can all be found in [@guillotIHES] together with a large amount of additional information. Yet the discussion conducted in Section 7 makes the article self-contained in the sense that proofs for the properties of these vector fields needed for Theorem C are provided. Additional motivation to conduct this discussion lies in the fact that the arguments provided are in line with the spirit of our method and differ from the original arguments of [@guillotIHES] and can be extended to encompass more general situations, see [@guillotOsaka]. Finally they lend themselves to a further new application: for $n=3$ they allow us to classify the first non-zero homogeneous component of a singularity of a complex vector field on a Kähler threefold, a result to be compared with Theorem C of [@ghreb]. Whereas this classification will not be carried out here, at the end of the paper we shall describe a step-by-step procedure to arrive to it.

[**Acknowledgements**]{}: The first author wants to express his gratitude to A. Guillot for many discussions concerning several objects present in this work and, in particular, for having explained to him several points of [@guillotIHES]. The second author is partially supported by Fundação para a Ciência e Tecnologia (FCT) through CMUP, through the Post-Doc grant SFRH/BPD/34596/2007 and through the project PTDC/MAT/103319/2008.

Overview of methods and results
===============================

This section contains a general description of the structure of the paper as well as some “qualitative” explanation of our techniques. “Quantitative” information required by the corresponding proofs will be supplied in the course of this paper. Some complements to the theorems stated in the Introduction will also be provided along with further comments on their contents.

First a point should be made about the vector fields and/or differential equations considered in this work. This is due to the fact that they are far from being “generic”. For example complete vector fields on ${{\mathbb C}}^n$ are very “non-generic” among polynomial/rational vector fields or yet among singular holomorphic foliations on projective spaces. Indeed the leaves of a foliation on ${{\mathbb C}}P(n)$ induced by a complete vector field are quotients of ${{\mathbb C}}$ as Riemann surfaces and this, by itself, is already very “non-generic”. Whereas they are “non-generic” their interest can hardly be questioned since, for example, they constitute a natural Lie algebra for the group of algebraic automorphisms of ${{\mathbb C}}^n$ and they remain an object intensively discussed as shown by the recent works of A. Bustinduy, L. Giraldo and others (cf. [@giraldo1], [@giraldo2] and their references). Actually in the study of differential equations we often encounter very special (i.e. “non-generic”) examples that turn out to play crucial roles in the theory. Apart from complete vector fields, our techniques also apply to semi-complete ones, i.e. to those vector fields admitting “maximal solutions” defined on subsets of ${{\mathbb C}}$ (cf. below). Halphen vector fields as studied in [@guillotIHES] satisfy this condition and they are discussed in Section 7. Their importance is clear since they appear in Mathematical Physics, in the theory of Ricci flow (namely the case of Ricci flow on homogeneous spaces) as well as in number theory through the celebrated functions $P, \, Q , \, R$ of Ramanujan. Yet another class of special equations/vector fields that fits in the pattern of our theory consists of those equations having meromorphic solutions defined on all of ${{\mathbb C}}$, in fact, they can be thought of as suitable semi-complete vector fields having meromorphic solutions defined on all of ${{\mathbb C}}$. These equations constitute a classical object of complex analysis and they include several well-known equations due to Painlevé and Chazy whose importance can hardly be overstated. This last direction of research however will not be exploited in this work though we hope it will be developed in the future. Another promising direction that is left for future investigation concerns the connections of our work with the point of view developed by X. Gomez-Mont and his collaborators, see [@xavier1], [@xavier2]. In fact, the study of some phenomena singled out in [@xavier2] by exploiting a simple “real” variant of our method seems to be very promising.

Let us now begin to outline the structure of this paper. Consider a polynomial vector field $X$ on ${{\mathbb C}}^n$ and denote by ${\mathcal{D}}$ the associated foliation induced on ${{\mathbb C}}P(n)$. Let $X_d$ stand for the top-degree homogeneous component of $X$ (having degree $d \geq 2$) and suppose that $X_d$ is not a multiple of the radial vector field. Under this assumption, the foliation ${\mathcal{D}}$ leaves the hyperplane at infinity $\Delta_{\infty}
= {{\mathbb C}}P(n) \setminus {{\mathbb C}}^n$ invariant. In addition, and modulo a minor remark discussed in Section 3, this foliation coincides with the foliation induced on $\Delta_{\infty}$ by $X_d$. The latter foliation can alternatively be seen as follows. The homogeneous character of $X_d$ implies that the direction of $X_d$ is invariant by homotheties of ${{\mathbb C}}^n$. Thus these directions have a well-defined projection on the quotient space of ${{\mathbb C}}^n$ by radial lines, i.e. on ${{\mathbb C}}P(n-1)$. Naturally we have $\Delta_{\infty} \simeq {{\mathbb C}}P(n-1)$ and the two foliations in question turn out to coincide under this identification. Furthermore they also coincide with the foliation induced on the exceptional divisor $\Delta_0 \simeq {{\mathbb C}}P(n-1)$ by the punctual blow-up of $X_d$ at the origin. The foliation associated to $X_d$ on ${{\mathbb C}}P (n)$ is going to be denoted by ${{\mathcal F}}$ and its restriction to $\Delta_{\infty}$ by ${{\mathcal F}}_{\infty}$. If $L_{\infty}$ is a leaf of ${{\mathcal F}}_{\infty}$ then the “cone over $L_{\infty}$” is invariant by ${{\mathcal F}}$.

Fundamentally our method consists of estimating the “speed” of the vector field $X$ near $\Delta_{\infty}$. This is done in two steps. First whereas $X$ has poles over $\Delta_{\infty}$, we can locally eliminate from $X$ the unbounded factor so as to obtain a “local regular vector field” about every regular point $p \in \Delta_{\infty}$ of ${{\mathcal F}}_{\infty}$. These “local vector fields” however do not patch together in a “foliated” global vector field since two representatives obtained through overlapping coordinates differ in general by a multiplicative constant. This means that they define, nonetheless, an affine structure (induced by $X_d$ or by $X$) on every leaf of ${{\mathcal F}}_{\infty}$. Versions of this affine structure already appeared in [@guillotFourier] and in some previous work of the first author under the name of “renormalized time-form”, it also plays an important role in [@guillotreb]. For the present work, the interest of this affine structure arises from the fact that it lends itself well to provide precise estimates of the flow of $X$ as long as precise estimates for the “distance” from the orbit in question to $\Delta_{\infty}$ are available.

Here it comes the second ingredient of our construction, namely a quantitative measure of “the rate of approximation” of a leaf of ${{\mathcal F}}$ to $\Delta_{\infty}$. Because $\Delta_{\infty} \subset {{\mathbb C}}P(n)$ and the Fubini-Study metric on ${{\mathbb C}}P(n)$ has positive curvature, it is well-known that complex submanifolds always bend themselves towards $\Delta_{\infty}$. In our case, this implies that the distance (relative to the Fubini-Study metric) of a leaf $L$ of ${{\mathcal F}}$ to $\Delta_{\infty}$ can never reach a local minimum unless this minimum is [*zero*]{}. The second ingredient of our method is reminiscent from this remark. Indeed we shall use the euclidean metric on suitably chosen affine coordinates, which is especially well-adapted to work with the above mentioned affine structure. Besides by exploiting the fact that we have leaves of an actual foliation (rather than general complex submanifolds) a quantitative version of the rate of approximation of a leaf to $\Delta_{\infty}$ is obtained. The phenomenon goes essentially as follows. At each regular point $p$ of a leaf $L$ of ${{\mathcal F}}$ we can single out the steepest descent direction of $L$ towards $\Delta_{\infty}$, namely the negative of the gradient of the distance function restricted to $L$. This yields a singular real one-dimensional oriented foliation ${\mathcal{H}}$ on $L$. Furthermore the conformal structure of $L$ is such that the foliation ${\mathcal{H}}^{\perp}$ orthogonal to ${\mathcal{H}}$ is constituted by level curves for the mentioned distance function. Roughly speaking an exponential rate of approximation for $L$ to $\Delta_{\infty}$ over the trajectories of ${\mathcal{H}}$ can then be derived. This estimate combines to the “uniform” estimates involved in the mentioned foliated affine structure to produce accurate estimates for the time taken by the flow of $X$ over trajectories of ${\mathcal{H}}$. In particular it is shown that the time taken by $X$ to cover an entire (infinite) trajectory is [*finite*]{} provided that the trajectory remains away from the singularities of ${{\mathcal F}}$ lying in $\Delta_{\infty}$. This results is then sharpened in Section 5 by allowing the trajectory to accumulate on (simple) singular points and still obtaining an analogous estimate. In particular there is only one special type of “simple” singularity that may yield an “endpoint” for the trajectories of ${\mathcal{H}}$ and, in this case, this will be an intersection point between the leaf $L$ and the hyperplane $\Delta_{\infty}$: the corresponding trajectory of ${\mathcal{H}}$ should then be thought of as being “finite”. Switching back and forward between estimates involving $X_d$ and estimates involving $X$ is not very complicated since $X$ is close to $X_d$ near $\Delta_{\infty}$.

The material mentioned above is covered in Sections 3, 4 and 5. Armed with these results we begin in Section 6 to prove the theorems stated in the Introduction. Theorems A and A’ are very natural. Since $X$ is complete its integral over a trajectory of ${\mathcal{H}}$ cannot converge. Besides this trajectory can never “reach” $\Delta_{\infty}$ since $X$ is complete on ${{\mathbb C}}^n$. This contrasts with our previous estimate asserting convergence of the integral in question provided that the corresponding trajectory of ${\mathcal{H}}$ remains away from the singularities of ${{\mathcal F}}$ (or ${\mathcal{D}}$) lying in $\Delta_{\infty}$. The only possible explanation for this apparent contradiction is that the flow of $X$ spend all but a finite amount of its existence on arbitrarily small neighborhoods of these singularities. The proof of Theorem A’ goes along similar ideas. In fact, the estimates carried over trajectories of ${\mathcal{H}}$ remain valid for every oriented foliation ${\mathcal{H}}^{\theta}$ forming an angle $\theta \in
(-\pi/2 , \pi/2)$ with ${\mathcal{H}}$. Once again the foliations ${\mathcal{H}}^{\theta}$ are well-defined thanks to the existence of a preferred conformal structure on the leaves of ${{\mathcal F}}, \, {\mathcal{D}}$. Modulo fixing a base point and using the obvious identifications, the union of all these trajectories span an unbounded region of ${{\mathbb C}}$ viewed as the domain of definition of the solution in question. The area of this region is “large” and, heuristically, can be imagined as “half of the area of large discs”.

These two theorems are clearly sharp in the sense that the leaves of ${\mathcal{D}}$, in general, can only confine themselves at singularities over certain special directions. In other words, we cannot have the whole complement of a compact part of a leaf being confined at a singularity. For example a simply connected leaf having a “punctual” end becomes a rational curve modulo adding the end to the leaf as it promptly follows from the classical Remmert-Stein theorem. Also simple complete polynomial vector fields such as $$y \frac{\partial}{\partial y} + xy \left[ x \frac{\partial}{\partial x} - y \frac{\partial}{\partial y} \right]$$ on ${{\mathbb C}}^2$ already exhibit non-properly embedded (simply connected) orbits accumulating on all of the “line at infinity” in the projective plane. Leaves that are “cylinders” are easier to handle as it is known since [@suzuki]. Yet they may have transcendent ends.

In view of Theorems A and A’, it is natural to imagine that the singular set of ${\mathcal{D}}$ contains significative information about the global geometry of complete polynomial vector fields. Theorem B is a contribution to the study of these vector fields as well as a test for the extent to which their global dynamics can be determined by the structure of their singularities. As mentioned the idea is to consider only “simple singularities” and to check what can be said about the vector field. From this point of view Theorem B is totally satisfactory since its statement is clearly sharp.

The proof of Theorem B is arguably the best application of our techniques and its main ingredients are going to be described in the sequel. The central difficulty is to guarantee the existence of a “dicritical singularity” for ${\mathcal{D}}$ in $\Delta_{\infty}$, i.e. a linearizable singularity all of whose eigenvalues belong to ${{\mathbb R}}_+$. The existence of this type of singularity implies in particular that the generic orbit of the vector field $X$ is of type ${{\mathbb C}}^{\ast}$ in the sense of [@suzuki] and several additional properties follow at once. To ensure the existence of this singularity is however a subtle question that can be approached as follows. First we replace $X$ by its top-degree homogeneous component $X_d$ along with its associated foliation denoted by ${{\mathcal F}}$. The property of having a dicritical singularity at $\Delta_{\infty}$ is common to ${\mathcal{D}}$ and ${{\mathcal F}}$ so that it is more convenient to work with a homogeneous vector field. Nonetheless we should take into account that $X_d$ is no longer complete but only [*semi-complete*]{}. Recall that a vector field $Z$ defined on an open set $U$ is said to be semi-complete (in $U$) if for every $p \in U$ there exists a solution $\phi : V \subset {{\mathbb C}}\rightarrow U$ of $X$, with $\phi (0) =p$, such that whenever a sequence $\{ T_i \}
\subset V$ converges to a point $\hat{T}$ in the boundary of $V$ the corresponding sequence $\phi (T_i)$ leaves every compact set in $U$. In this sense $V$ is a maximal domain of definition for the solution $\phi$ of $Z$. Being only semi-complete, it may happen that $X_d$ “reaches the infinity in finite time” and this gives rise to some additional difficulties.

An important difficulty arising from the difference between complete and semi-complete vector fields lies in the fact that the leaves of the foliation associated to a semi-complete vector field may be hyperbolic Riemann surfaces. An example of this phenomenon is, indeed, provided by Halphen vector fields, cf. Section 7. However, in the case of the foliation ${{\mathcal F}}$ associated to $X_d$ it can be proved that the corresponding leaves are still quotients of ${{\mathbb C}}$. This is done by resorting to a result due to Brunella concerning the pluri-subharmonic variation of the foliated Poincaré metric, cf. [@marco2]. The solutions of $X_d$ will therefore be meromorphic functions defined on ${{\mathbb C}}$ or in ${{\mathbb C}}$ minus a point. Next we bring in our results involving the time taken by $X_d$ to cover trajectories of ${\mathcal{H}}$ in the singular context (here it is used the main result of Section 5, namely Theorem \[maintheo\]). Theorem \[maintheo\] immediately implies that the solutions cannot be meromorphic on all of ${{\mathbb C}}$ and, by exploiting some additional properties of the foliations ${\mathcal{H}}, \, {\mathcal{H}}^{\perp}$, a contradiction ensuring the existence of the desired dicritical singularity is finally obtained.

Let us now make some comments about the assumption that the singularities of ${\mathcal{D}}$ lying in $\Delta_{\infty}$ are simple. First a reader more familiar with complete polynomial vector fields might be tempted to think that the condition imposed on the singularities of ${\mathcal{D}}$ immediately implies that the vector field $X$ must have degree $2$. This is however not the case. For example some of these singularities may be a codimension 1 saddle-node singularity (as in item 2 of the definition of simple singularities given in the Introduction). Also the statement of Theorem B may be extended to encompass fair more general singularities among the very general class of “absolutely isolated singularities”, cf. [@canoetc]. While we shall not seek to accurately establish any of these extensions, at the very end of Section 5 the reader will find some information on the structure of more degenerate singularities for which our methods will still work. It is also interesting to observe that our techniques apply equally well to rational vector fields and not only to polynomial ones. In practice this change the “expected” order of $\Delta_{\infty}$ as component of the “pole divisor” of the vector field in question and leads to numerous additional possibilities whose analysis may partially be facilitated by our ideas. In particular several Painlevé equations fall in this class of problems.

As mentioned A. Bustinduy, L. Giraldo and their collaborators have been investigating properties of the solutions of complete vector fields through various methods such as the theory of Nevanlinna and Andersen-Lempert theories, see [@giraldo1], [@giraldo1] and their references. Similarly, if take into account a classical result due to Forstneric [@forst], our method is likely to find some applications in the theory of holomorphic differential equations blowing-up in finite real time. This should lead to some progress in questions similar to those treated by Fornaess and Grellier in [@fornaess] which itself connects with previous works by a number of authors including possible applications in the spirit of [@fornaessbuzzard].

Finally in view of the beautiful results obtained by A. Guillot in [@guillotFourier], [@guillotIHES], it is natural to try to apply our methods to quadratic semi-complete vector fields as those considered in the mentioned works. The material prepared for the proofs of Theorems A, A’ and B allows us to show that one such vector field having no dicritical singularity at infinity has leaves that are hyperbolic Riemann surfaces. This quickly leads to Theorem C by resorting again to Brunella’s result on the variation of the Poincaré metric, see [@marco2]. Halphen vector fields are however very special in the sense that they [*do have*]{} dicritical singularities at infinity and still the leaves of their associated foliation may be hyperbolic Riemann surfaces. Whereas these results, and many others, are due to A. Guillot and appear in [@guillotIHES], we found it was worth re-obtaining them by following our general point of view. This discussion takes up most of the last section of this paper. In particular it involves some considerations about convergence of Poincaré series that differ from their classical theory.

Another motivation for us to revisit Guillot’s work on Halphen vector fields is to pave the way for one further application of our method. As briefly stated in the Introduction, the arguments developed in our discussion about Halphen vector fields can be adapted to provide a classification of the first homogeneous components at a singular point of a globally defined holomorphic vector field on a compact Kähler threefold. In addition the same classification applies equally well to the top degree homogeneous component of a complete polynomial vector field on ${{\mathbb C}}^3$. However, unlike the other results of this paper, our techniques does not allow to derive a similar classification for dimensions higher than $3$. In fact, dimension $3$ turns out to be special for two reasons: on one hand the groups of automorphisms of compact Riemann surfaces are easy to describe and, on the other hand, the existence of invariant surfaces for the corresponding foliations enables us to apply a powerful theorem due to McQuillan which is available only in dimension $2$. Details on this procedure are to be found in Section 7.3.

To close this discussion we would like to point out a curious remark involving Theorem C and, more generally, semi-complete homogeneous vector fields. In fact singularities of homogeneous vector field on ${{\mathbb C}}^3$ (or ${{\mathbb C}}^n$) possess a natural meromorphic “dual” represented by a neighborhood of the (hyper-) plane at infinity (even though this neighborhood cannot be collapsed to a singular point). More precisely, as detailed in Section 3, the blow-up at the origin of a homogeneous vector field leads to an exceptional divisor sharing a natural “duality” with the divisor obtained at infinity of the corresponding projective space. We shall refer to a neighborhood of the hyperplane at infinity as the [*dual singularity*]{} (assuming that the singularity of a homogeneous polynomial vector field is implicitly fixed). By virtue of Theorem C and of the global realization of Halphen vector fields constructed in [@guillotIHES], it is natural to ask whether the dual singularity of a hyperbolic Halphen vector field can be realized by a complete meromorphic vector field on a complex $3$-manifold not necessarily compact (where by complete meromorphic vector field it is meant a meromorphic vector field that is complete in the complement of its pole locus). The answer to this question turns out to be negative as it follows from the discussion in Section 7.

Homogeneous vector fields and their foliations {#sechom}
==============================================

Unless otherwise stated, throughout this paper all homogeneous vector fields have degree $d \geq 2$ and are supposed not to be a multiple of the radial vector field. In this section we shall work in dimension $3$ rather than in ${{\mathbb C}}^n$ just to abridge notations since all arguments presented in the sequel can be carried over word-by-word to higher dimensions.

Consider a homogeneous polynomial vector field $X$ of degree $d \geq 2$ defined on ${{\mathbb C}}^3$. Since $X$ is homogeneous, its associated foliation ${{\mathcal F}}$ is invariant by homotheties of the form $(x, y, z) \mapsto (\lambda x,\lambda y ,\lambda z)$, $\lambda \in {{\mathbb C}}^{\ast}$ and, therefore, also induces a foliation on ${{\mathbb C \mathbb P}}(2)$. An alternative way to look at this situation consists of punctually blowing-up $X$ at the origin of ${{\mathbb C}}^3$. We denote by ${{\widetilde{\mathbb C}}}^3$ the corresponding blow-up of ${{\mathbb C}}^3$ and by $\Delta_0 =\pi^{-1} (0)$ the resulting exceptional divisor, where $\pi: {{\widetilde{\mathbb C}}}^3 \mapsto {{\mathbb C}}^3$ represents the punctual blow-up of ${{\mathbb C}}^3$ at the origin. The transform (blow-up) ${{\widetilde{X}}}$ (resp. ${{\widetilde{\mathcal F}}}$) of $X$ (resp. ${{\mathcal F}}$) vanishes identically over $\Delta_0$ (resp. leaves $\Delta_0$ invariant), as it follows from the fact that the degree of $X$ is strictly greater than $1$ (resp. that $X$ is not a multiple of the radial vector field).

Recalling also that ${{\widetilde{\mathbb C}}}^3$ can be viewed as a line bundle over $\Delta_0 =\pi^{-1}
(0)$, let ${{\mathcal P}}_0$ denote the bundle projection ${{\mathcal P}}_0: {{\widetilde{\mathbb C}}}^3
\rightarrow \Delta_0$. This line bundle can be compactified into a projective line bundle by adding the “section at infinity” $\Delta_{\infty}$. Denoting by $M$ the total space of the resulting projective line bundle, it follows that $M$ comes with two bundle projections ${{\mathcal P}}_0, \, {{\mathcal P}}_{\infty}$ realizing it as a projective bundle respectively over $\Delta_0, \, \Delta_{\infty}$. The manifold $M$ is also isomorphic to the blow-up of ${{\mathbb C \mathbb P}}(3)$ at the origin. The vector field ${{\widetilde{X}}}$ can meromorphically be extended to $M$ so that it induces a holomorphic foliation, still denoted by ${{\widetilde{\mathcal F}}}$, on all of $M$. Besides ${{\widetilde{\mathcal F}}}$ leaves both $\Delta_0, \, \Delta_{\infty}$ invariant since $X$ is homogeneous and it is not a multiple of the radial vector field. The foliation induced on $\Delta_0$ (resp. $\Delta_{\infty}$) by restriction of ${{\widetilde{\mathcal F}}}$ is going to be denoted by ${{\widetilde{\mathcal F}}}_0$ (resp. ${{\widetilde{\mathcal F}}}_{\infty}$). Because ${{\widetilde{\mathcal F}}}$ comes from a homogeneous vector field, these foliations coincide with the restrictions of ${{\widetilde{\mathcal F}}}$ to $\Delta_0, \, \Delta_{\infty}$. As to the vector field ${{\widetilde{X}}}$, its pole divisor coincides with $\Delta_{\infty}$ and it has order $d-1>0$. The zero divisor of ${{\widetilde{X}}}$ is the union of $\Delta_0$ (a component of order $d-1 >0$) with the transform of the zero divisor of $X$.

Because our foliations are singular, we shall consider a definition of “regular leaf” allowing, in some cases, a leaf to “go through” a singularity. For some applications, it may be of interest to use the definition given in [@marco2] which we briefly recall for the convenience of the reader. Consider the $n$-dimensional polydisc ${\bf D}^n$ and the trivial fibration ${\bf D}^n = {\bf D}^{n-1} \times {\bf D} \rightarrow {\bf
D}^{n-1}$. A meromorphic map $f : {\bf D}^n \rightarrow M$ is said to be a [*foliated meromorphic immersion*]{} if the following conditions are verified.

1\. The indeterminacy set $I (f)$ of $f$ intersects each vertical fiber of ${\bf D}^n$ over a discrete set.

2\. $f$ is an immersion on the complement of $I (f)$ and, in this complement, takes vertical fibers to leaves of ${{\widetilde{\mathcal F}}}$ (more generally of the foliation under consideration).

Consider a regular point $p$ in $M \setminus {\rm Sing}\, ({{\widetilde{\mathcal F}}})$. Restricted to $M \setminus {\rm Sing}\, ({{\widetilde{\mathcal F}}})$ the foliation ${{\widetilde{\mathcal F}}}$ is regular so that the leaf $L_p$ through $p$ has an obvious sense. A closed subset $K \subset L_p$ is called a [*vanishing end*]{} of $L_p$ if all the conditions below are satisfied:

1\. $K$ is isomorphic to the punctured disc and the holonomy of the restriction of ${{\widetilde{\mathcal F}}}$ to $M \setminus {\rm Sing}\,
({{\widetilde{\mathcal F}}})$ corresponding to the cycle $\partial K$ has finite order $k$.

2\. There is a foliated meromorphic immersion $f : {\bf D}^n \rightarrow M$ such that

2.a

:   $I (f) \cap (\{ 0\} \times {\bf D}) =  0 \in {\bf D} \subset {{\mathbb C}}$, where $\{ 0 \}$ stands for the origin of ${\bf D}^{n-1} \subset {{\mathbb C}}^{n-1}$.

2.b

:   The image of $f$ restricted to $(\{ 0\} \times {\bf D})$ is the interior of $K$. Furthermore $f: (\{ 0\} \times {\bf D})
    \rightarrow {\rm Int} \, (K)$ is a regular covering of degree $k$, where ${\rm Int} \, (K)$ stands for the interior of $K$.

Given a regular point $p \in M \setminus {\rm Sing}\, ({{\widetilde{\mathcal F}}})$, consider the regular leaf $L_p$ through $p$ relative to the restriction of ${{\widetilde{\mathcal F}}}$ to $M \setminus {\rm Sing}\, ({{\widetilde{\mathcal F}}})$. If $L_p$ possesses no vanishing ends, then the regular leaf of ${{\widetilde{\mathcal F}}}$ containing $p$ is exactly $L_p$. Otherwise this leaf will consist of $L_p$ with the ends of the vanishing ends added to it. Naturally when no misunderstanding is possible, this leaf will still be denoted by $L_p$. Here we observe that the operation of adding an end to $L_p$ should be understood in the sense of orbifolds: the multiplicity of the added point will precisely be the order $k$ of the holonomy relative to $\partial K$. Naturally such orbifolds can [*a posteriori*]{} be made into Riemann surfaces by standard normalization.

Let us now continue with the discussion of the structure of the foliation ${{\widetilde{\mathcal F}}}$ over $M$. Naturally the singular set of ${{\widetilde{\mathcal F}}}$ has codimension at least $2$. Besides this singular set is saturated (i.e. invariant) by the fibers of ${{\mathcal P}}_0$ (resp. ${{\mathcal P}}_{\infty}$) due to the invariance of ${{\mathcal F}}$ by homotheties of the form $(x, y, z) \mapsto (\lambda x,
\lambda y, \lambda z)$, $\lambda \in {{\mathbb C}}^{\ast}$. In particular, the foliations ${{\widetilde{\mathcal F}}}_0, \, {{\widetilde{\mathcal F}}}_{\infty}$ automatically have singular sets of codimension at least $2$ [*inside $\Delta_0, \, \Delta_{\infty}$*]{} (in other words the intersection of the singular set of ${{\widetilde{\mathcal F}}}$ with ${{\widetilde{\mathcal F}}}_0, \, {{\widetilde{\mathcal F}}}_{\infty}$ yields a set of codimension at least $2$ inside $\Delta_0, \, \Delta_{\infty}$).

Consider a non-algebraic leaf $L$ of ${{\widetilde{\mathcal F}}}$ not contained in $\Delta_0 \cup \Delta_{\infty}$. The projection of $L$ onto $\Delta_0$ (resp. $\Delta_{\infty}$), ${{\mathcal P}}_0 (L) =L_0$ (resp. ${{\mathcal P}}_{\infty} (L) =L_{\infty}$), is clearly a leaf of ${{\widetilde{\mathcal F}}}_0$ (resp. ${{\widetilde{\mathcal F}}}_{\infty}$) since the initial vector field $X$ is homogeneous. Furthermore one immediately checks that the restriction of ${{\mathcal P}}_0$ (resp. ${{\mathcal P}}_{\infty}$) to $L$ realizes $L$ as an Abelian covering of $L_0$ (resp. $L_{\infty}$). Therefore we conclude:

1.  The non-compact leaves $L, \, L_0, \, L_{\infty}$ have all the same nature: either they are covered by ${{\mathbb C}}$ or they are covered by the unit disc $D$.

2.  $L_0, \, L_{\infty}$ are isomorphic as Riemann surfaces. $L$ is an Abelian covering of $L_0, \, L_{\infty}$.

In this way, we may focus on the behavior of ${{\widetilde{X}}}$ near its pole divisor $\Delta_{\infty}$ or near $\Delta_0$ according to our convenience. Next consider a leaf $L_{\infty}$ of ${{\widetilde{\mathcal F}}}_{\infty}$. By [*the cone over $L_{\infty}$*]{} it is meant the $2$-dimensional immersed singular surface ${{\mathcal P}}_{\infty}^{-1} (L_{\infty})$ which is invariant by ${{\widetilde{\mathcal F}}}$. In other words, if $\psi(T) = (x(T), y(T),0)$, $T \in \Omega \subseteq {{\mathbb C}}$, is a local parametrization of $L_{\infty}$, then the cone is parameterized by $\Phi(T,z) = (x(T),
y(T),z)$, $z \in {{\mathbb C}}$. The singular points of ${{\mathcal P}}_{\infty}^{-1}
(L_{\infty})$ belong to fibers sitting over the singular set of ${{\widetilde{\mathcal F}}}_{\infty}$ which, we recall, may intersect $L_{\infty}$ non-trivially due to the above given definition of “regular leaf”. Away from its singularities, ${{\mathcal P}}_{\infty}^{-1} (L_{\infty})$ can be viewed as a complex surface equipped with a singular holomorphic foliation. Let us then denote by $S$ this surface and by ${{\widetilde{\mathcal F}}}_S$ the foliation on $S$ obtained by restriction of ${{\widetilde{\mathcal F}}}$ to $S$. Note that $S$ is invariant under the automorphism $(x,y,z)\mapsto
(x,y,{\lambda}z)$, ${\lambda}\in {{\mathbb C}}^*$ and so is the foliation ${{\widetilde{\mathcal F}}}_S$.

Since $S$ is a 2-dimensional variety, ${{\widetilde{\mathcal F}}}_S$ is a codimension $1$ singular foliation on $S$ and hence it is transversely conformal. This allows us to keep good control of the directions over which the leaves of ${{\widetilde{\mathcal F}}}_S$ “become closer one to the others” modulo choosing an auxiliary Hermitian metric. The idea is well-known and can be found, for instance, in [@Ghys-bourb]. In our case however, we shall use an explicit parametrization. For this, let $M$ be equipped with affine coordinates $(x , y, z)$ on $M$ such that

1.  $\{ z=0\} \subset \Delta_{\infty}$, $(x , y) \in
    {{\mathbb C}}^2$, $z \in {{\mathbb C}}$.

2.  the vector field ${{\widetilde{X}}}$ is given by $${{\widetilde{X}}}= \frac{1}{z^{d-1}} \left[ F (x , y) \frac{\partial}{\partial
    x} + G (x , y) \frac{\partial}{\partial y} + zH (x , y)
    \frac{\partial}{\partial z} \right]   \label{tXX}$$ where $F ,G$ are polynomials of degree either $d$ or $d+1$ and $H$ is a polynomial of degree $d$ (the independence of $F ,G$ and $H$ on $z$ is a consequence of the homogeneity of $X$).

3.  The projection ${{\mathcal P}}_{\infty} : M \rightarrow \Delta_{\infty}$ in the above coordinates becomes $(x , y, z) \mapsto (x , y)$.

Note that $\Delta_{\infty}$ is itself isomorphic to ${{\mathbb C \mathbb P}}(2)$. In particular the affine coordinates $(x ,y)$ for $\Delta_{\infty}$ defines a notion of “line at infinity” for $\Delta_{\infty}$ itself. We shall denote this “line” by $\Delta_{\infty}^{(x,y)}$. In particular we note that the domain of definition of the coordinates $(x , y, z)$ coincides with $M \setminus (\Delta_0 \cup {{\mathcal P}}_{\infty}^{-1} (\Delta_{\infty}^{(x,y)}))$. Naturally the choice of the affine coordinates $(x,y)$ and of the line $\Delta_{\infty}^{(x,y)}$ are not canonical. For a generic choice of these coordinates, $\Delta_{\infty}^{(x,y)}$ does not contain singular points of the corresponding foliation on $\Delta_{\infty}$ and, besides, $\Delta_{\infty}^{(x,y)}$ is not invariant by this foliation. Now we have:

\[lineatinfinitygeneric\] Suppose that the affine coordinates $(x,y)$ are chosen so that the resulting “line at infinity” $\Delta_{\infty}^{(x,y)}$ is not invariant by the corresponding foliation on $\Delta_{\infty}$. Then the top-degree component of the vector field ${{\widetilde{X}}}$ has the form $$\label{topdegree}
f(x , y) [x \partial /\partial x + y \partial /\partial y + z {\partial /\partial z}]$$ for a degree $d$ homogeneous polynomial $f$.

. Suppose that the initial homogeneous vector field $X$ is given by $X = A {\partial /\partial x}+ B {\partial /\partial y}+ C{\partial /\partial z}$. Then, with the above notations, ${{\widetilde{X}}}= z^{1-d} [F \partial /\partial x
+ G \partial /\partial y + zH \partial /\partial z]$ where $F(x,y) = A(x,y,1) - x C(x,y,1)$, $G(x,y) = B(x,y,1) - y C(x,y,1)$ and $H(x,y) = - C(x,y,1)$. Thus the assumption on the degree of $F$ (resp. $G, \, H$) follows from the fact that we can assume $A$ (resp. $B, \, C$) not divisible by $z$. This also implies that the top degree of $F$ (resp. $G, \, H$) is given by $x$ (resp. $y, \, z$) times the top degree of $C$, i.e. the top degree component of ${{\widetilde{X}}}$ has the form (\[topdegree\]).

A further comment concerning the difference between the foliation ${{\widetilde{\mathcal F}}}_{\infty}$ induced by ${{\widetilde{X}}}$ on $\Delta_{\infty}$ and the corresponding foliation ${{\widetilde{\mathcal F}}}$ in the $3$-dimensional space is also needed. To be more precise, consider the vector field ${{\widetilde{X}}}$ given by Formula (\[tXX\]) in the coordinates $(x,y,z)$. If $F,G$ have only trivial common factors, then the foliation induced by ${{\widetilde{X}}}$ on $\Delta_{\infty}$ is given in $(x,y, \{z=0 \})$ coordinates by $F \partial /\partial x + G  \partial /\partial y$. Suppose now that $F$ and $G$ possess nontrivial common factors. Set $\textsc{P} =
{\rm g.c.d.}\, (F,G)$ so that $F = \textsc{P}. a(x,y)$ and $G =
\textsc{P} . b(x,y)$ with $a,b$ having only trivial common factors. In this case the foliation ${{\widetilde{\mathcal F}}}_{\infty}$ is actually represented by the vector field $a(x,y) \partial /\partial x + b(x,y) \partial
/\partial y$. If $\textsc{P}$ also divides $H$, then this foliation still coincides with the restriction of ${{\widetilde{\mathcal F}}}$ to $\Delta_{\infty}$. Thus the only relevant case occurs when $\textsc{P}$ does not divide $H$. After eliminating all common factors, we can suppose that ${\rm g.c.d.}\, (\textsc{P}, H)$ is invertible. Summarizing this discussion, we see that the foliation associated to ${{\widetilde{X}}}$ is given by a polynomial vector field of the form $$Y = \textsc{P} \left[ a (x,y) \frac{\partial}{\partial
x} + b (x , y) \frac{\partial}{\partial y} \right] + z H (x , y)
\frac{\partial}{\partial z} \, ,$$ where ${\rm g.c.d.}\, (\textsc{P}, H)$ is constant. It follows in particular that the projective curve $\{ \textsc{P}
=0 \} \subset \Delta_{\infty}$ (if not empty) is constituted by singularities of ${{\widetilde{\mathcal F}}}$ whereas its “generic” point is regular for ${{\widetilde{\mathcal F}}}_{\infty}$. Besides there are two different possibilities that need to be considered:

(a)

:   $\{ \textsc{P} = 0 \} \subset \Delta_{\infty}$ is invariant by ${{\widetilde{\mathcal F}}}_{\infty}$.

(b)

:   $\{ \textsc{P} =0 \} \subset \Delta_{\infty}$ is not invariant by ${{\widetilde{\mathcal F}}}_{\infty}$.

\[willitbeuseful?\] [In much of what follows the possibility of having $\{ \textsc{P} = 0 \}$ invariant by ${{\widetilde{\mathcal F}}}_{\infty}$ will be excluded from the discussion since it does not take place in the case of semi-complete (and hence of complete) vector fields, cf. Proposition \[aproposition\].]{}

Our purpose is now to equip the leaves of ${{\widetilde{\mathcal F}}}$ in $\Delta_{\infty}$ with an abelian form $\omega_1$ naturally related to the holonomy of the leaf in question. This will be done in the affine ${{\mathbb C}}^3$ where the coordinates $(x,y,z)$ are defined. With the preceding notations, let us fix a regular leaf $L_{\infty} \subset \Delta_{\infty}$ and a point $p \in L_{\infty}$. Assume that $p$ is a regular point for ${{\widetilde{\mathcal F}}}$ or that $p$ is a singular point for the same foliation being $L_{\infty}$ a connected separatrix for ${{\widetilde{\mathcal F}}}$ at $p$. Under this assumption, the leaf $L_{\infty}$ can locally be parametrized in the form $(x,y(x))$, or $(x(y),y)$, and $z=0$. It suffices to consider a local parametrization of the form $(x,y(x))$ since the other possibility is analogous. The vector field ${{\widetilde{X}}}$ then yields $$\frac{dz}{dx} = \frac{H(x,y(x))}{F(x,y(x))} z \; .$$ Therefore $$z = z_0 \exp \left[ \int_{x_0}^x \frac{H(x,y(x))}{F(x,y(x))} dx \right] \; . \label{omega1}$$ Thus we define an abelian form $\omega_1$ on $L_{\infty}$ by declaring that the coefficient of $\omega_1$ at $(x,y(x))$ is nothing but $-H(x,y(x)) / F(x,y(x))$ (the minus sign is only a matter of convention). In particular we note that possible non-trivial common factors between $F,H$ are automatically canceled out in the definition of $\omega_1$. If the leaf were parameterized in the form $(x(y),y)$, the analogous result would yield for coefficient $-H(x,y(x)) / G(x,y(x))$. The form $\omega_1$ is the “logarithmic derivative of the holonomy” for the foliation ${{\widetilde{\mathcal F}}}_S$ induced on the cone $S$ over $L_{\infty}$. This means the following: let $L$ be a leaf of ${{\widetilde{\mathcal F}}}_S$ and consider a path $c:[0,1] \mapsto
L$, on $L$. Denoting by ${\rm Hol}(c)$ the holonomy associated to $c$, we have $$({\rm Hol} (c))^{\prime}(c(0)) = e^{-\int_c \omega_1} \, ,$$ where ${\rm Hol} (c)$ is identified with a map between open sets of ${{\mathbb C}}$ equipped with the coordinate $z$.

Fixed a regular leaf $L_{\infty} \subseteq \Delta_{\infty}$ of ${{\widetilde{\mathcal F}}}$ there are real trajectories, or paths, contained in $L_{\infty}$ and possessing a contractive holonomy. To construct these trajectories we proceed as follows. The Abelian form $\omega_1$ induces on $L_{\infty}$ a pair of real 1-dimensional oriented singular foliations: the foliations given by $\{ {\rm
Im}(\omega_1) = 0\}$ and by $\{ {\rm Re}(\omega_1) = 0\}$. Denote by $\mathcal{H}$ the oriented foliation defined by $\{ {\rm
Im}(\omega_1) = 0\}$, being the orientation determined by the positivity of ${\rm Re}(\omega_1)$, i.e. if $\phi(t)$ is a parametrization of a leaf of $\mathcal{H}$ then ${\rm Re}(\omega_1.\phi^{\prime}(t))
= \omega_1.\phi^{\prime}(t) > 0$. Each oriented trajectory of the foliation $\mathcal{H}$ will be called a real trajectory.

It is clearly necessary to have information about the singular set of $\mathcal{H}$. Since $\mathcal{H}$ depends only on the foliation associated to ${{\widetilde{X}}}$ (rather than of ${{\widetilde{X}}}$ itself), we identify four “critical regions” (possibly) giving rise to singularities for $\mathcal{H}$, namely:

1.  Singular points of ${{\widetilde{\mathcal F}}}_{\infty}$.

2.  Points in the curve $\{ H =0 \}$ (assuming, as already done, that ${\rm g.c.d.}\, (\textsc{P}, H)$ is a constant).

3.  Points in the curve $\{ \textsc{P} =0 \}$.

4.  The line at infinity $\Delta_{\infty}^{(x,y)} \subset \Delta_{\infty}$ (defined by means of the affine coordinates $(x,y)$).

In the sequel we shall determine the structure of the foliation ${\mathcal{H}}$ in cases 2, 3 and 4 above. The discussion of singular points of ${{\widetilde{\mathcal F}}}_{\infty}$ will mostly be carried out in Section 4. Note that points belonging simultaneously to $\{ H =0 \}$ and to $\{ \textsc{P} =0 \}$ are somehow “more degenerate” and they might require a more detailed analysis. Yet these points are in finite number so that for global dynamics considerations as those carried out in the subsequent sections they can be ignored. Actually in most of our applications we shall deal with the case where $\textsc{P}$ is constant. As to points belonging to the line at infinity, they will all be treated with the same argument whether or not they belong to the (closures of) $\{ H =0 \}$ or to $\{ \textsc{P} =0 \}$. Finally, concerning the case (4) above, it naturally depends on the choice of the affine coordinates $(x,y)$. A generic choice of these leads to a line at infinity $\Delta_{\infty}^{(x,y)}$ that neither is invariant by ${{\widetilde{\mathcal F}}}_{\infty}$ nor it contains singularities of it.

Let us begin with the curve $\{ H =0 \}$ corresponding to zeros of $\omega_1$.

\[tiposing\] Let $p \in \Delta_{\infty}$ be a regular point of ${{\widetilde{\mathcal F}}}$. Assume that $p$ lies in the curve $\{ H =0 \} \cap \Delta_{\infty}$ (but not in $\{ \textsc{P} =0 \}$). Then $p$ is a singular point for $\mathcal{H}$. Besides the local structure of $\mathcal{H}$ restricted to the leaf of ${{\widetilde{\mathcal F}}}$ through $p$ is a saddle with $2m$ (real) separatrices (for $m \geq 2$).

Consider local coordinates $(u,v,w)$ about $p$ where the leaf $L_p$ of ${{\widetilde{\mathcal F}}}$ through $p$ locally coincides with $\{ v=w=0\}$. Also we can suppose that $F (p) \neq 0$ modulo substituting $F$ by $G$. In fact, otherwise either $p$ would be a singular point of ${{\widetilde{\mathcal F}}}$ or a regular point of ${{\widetilde{\mathcal F}}}$ lying in the curve $\{ \textsc{P} =0 \}$. As already mentioned, these cases will be treated later.

We then conclude that the restriction of $\omega_1$ to $L_p$ is a holomorphic $1$-form about $p$ with a zero at $p$. The structure of the real foliation induced near a zero of a holomorphic $1$-form on a Riemann surface is always a saddle as in the statement. Here $m$ is precisely the order of $p$ as zero of $\omega_1$.

Let us now work out the behavior of ${\mathcal{H}}$ at points of $\{ \textsc{P} =0 \}$. Clearly it is sufficient to consider the domain of definition of the coordinates $(x,y,z)$. Let $\textsc{P} = \textsc{P}_1^{k_1} \cdots \textsc{P}_l^{k_l}$ be the decomposition of $\textsc{P}$ into irreducible components along with their corresponding multiplicity. We just need to consider the curve $\{ \textsc{P}_1^{k_1} =0 \}$.

\[curvaP=0\] Suppose that $\{ \textsc{P}_1 =0 \}$ is not invariant by ${{\widetilde{\mathcal F}}}_{\infty}$. If $k_1 \geq 2$ then $\omega_1$ has a pole of order $k_1 \geq 2$ at a generic point $p$ of this curve so that ${\mathcal{H}}$ has a saddle-singularity at $p$. On the other hand, if $k_1=1$, then $p$ is a simple pole for $\omega_1$ whose residue (not necessarily real) equals $H(p)/ F^{\ast} (p)$ where $F^{\ast} = F/  \textsc{P}_1$.

Again it is enough to consider generic points where $\{ \textsc{P}_1 =0 \}$ is transverse to ${{\widetilde{\mathcal F}}}_{\infty}$. At a generic point $p$ of $\{ \textsc{P}_1 =0 \}$ we have that $F^{\ast}(p) \ne 0$. Moreover, since we are assuming ${\rm g.c.d.}\, (\textsc{P}, H)$ to be invertible, we can also assume that $H(p) \ne 0$. The $1$-form $\omega_1$ has therefore a pole of order $k_1$ whose coefficient is equal to $H(p)/F^{\ast}$.

Let us now consider points belonging to the line at infinity $\Delta_{\infty}^{(x,y)} \subset \Delta_{\infty}$.

\[lineatinfinity\] Points belonging to $\Delta_{\infty}^{(x,y)}$ yield source singularities for ${\mathcal{H}}$ provided that the coordinates $(x,y)$ are generically chosen.

As already seen, $\Delta_{\infty}^{(x,y)}$ does not contain singular points of ${{\widetilde{\mathcal F}}}_{\infty}$. Besides $\Delta_{\infty}^{(x,y)}$ is not invariant by ${{\widetilde{\mathcal F}}}_{\infty}$ either. It then follows that each point $p$ in $\Delta_{\infty}^{(x,y)}$ locally belongs to a unique leaf $L_p$ of ${{\widetilde{\mathcal F}}}_{\infty}$ (by virtue of the definition of leaf given earlier in this section, this is not necessarily true if $\Delta_{\infty}^{(x,y)}$ is invariant by, or contains singular points of, ${{\widetilde{\mathcal F}}}_{\infty}$). Thus the map that assigning to $p$ the residue at $p$ of the $1$-form $\omega_1$ is globally defined on $\Delta_{\infty}^{(x,y)}$. It is lack of continuity is associated to points of tangency between $\Delta_{\infty}^{(x,y)}$ and ${{\widetilde{\mathcal F}}}_{\infty}$ since, on a neighborhood of these, a same leaf of ${{\widetilde{\mathcal F}}}$ intersects $\Delta_{\infty}^{(x,y)}$ more than once. Yet to establish the statement it suffices to check this residue equals $1$ at a generic point of $\Delta_{\infty}^{(x,y)}$. Indeed, this will imply that the residue must be real strictly positive at every point $p$ in $\Delta_{\infty}^{(x,y)}$ so that all these points constitute source singularities for ${\mathcal{H}}$.

Let us then consider those points where $\Delta_{\infty}^{(x,y)}$ is transverse to ${{\widetilde{\mathcal F}}}_{\infty}$. Let $(u,v,w)$ be new local affine coordinates for $M$ where $w$ is the coordinate transverse to $\Delta_{\infty}$ and such that the line at infinity $\Delta_{\infty}^{(x,y)}$ is given by $\{u = 0\}$. The standard change of coordinates associated is then given by $(u,v,w) \longmapsto (1/u, v/u, w) = (x,y,z)$. In these new coordinates, the vector field ${{\widetilde{X}}}$ becomes (up to multiplication by $w^{1-d}$) $$-u^2 F(1/u, v/u) \frac{\partial}{\partial u} + u(-vF(1/u, v/u) + G(1/u, v/u)) \frac{\partial}{\partial v}
+ w H(1/u, v/u) \frac{\partial}{\partial w} \; .$$ Recall that the polynomial vector field $F(x,y) \partial /\partial x + G(x,y) \partial /\partial y$ has degree $d+1$. Furthermore its component of degree $d+1$ has the form $f(x , y) [x \partial
/\partial x + y \partial /\partial y]$ where $f$ is homogeneous of degree $d$. In particular $u^2 F(1/u, v/u)$ has a pole of order $d-1$ over $\{ u=0\}$. Similarly the top-degree homogeneous component of $-vF(1/u, v/u) + G(1/u, v/u)$ vanishes identically so that $\Delta_{\infty}^{(x,y)}$ represents a polar component of degree $d-1$ for the component of ${{\widetilde{X}}}$ in the $v$-direction as well. Finally the order of poles of $H(1/u,v/u)$ over $\Delta_{\infty}^{(x,y)}$ equals $d$. Formula (\[omega1\]) then shows that $\omega_1$ has poles of order $1$ over $\Delta_{\infty}^{(x,y)}$. Indeed the principal part of $\omega_1$ is simply $1/u$, since the top degree component is given by Equation (\[topdegree\]). The statement follows.

To close this section, let us introduce the global notion of trajectory for the foliation $\mathcal{H}$ under the condition that the trajectory in question remains away from the singular set of ${{\widetilde{\mathcal F}}}$. It also convenient to consider the standard flat metric of ${{\mathbb C}}^3$ where the initial affine coordinates $(x,y,z)$ are defined. Roughly speaking a real trajectory is said to be [*finite*]{} if its length, with respect to the above metric, is finite. A precise definition of this length requires to define what are the [*endpoints of a trajectory*]{} which will be done in the sequel, cf. Remark \[rem\] below. To motivate this definition let $l$ be a real trajectory of $\mathcal{H}$ and let $c:[0,1]
\rightarrow l$ be a parametrization of the segment of this trajectory joining $p_0 = c(0)$ to $p_1 = c(1)$. Then the holonomy map ${\rm
Hol}(c): \Sigma_0 \rightarrow \Sigma_1$, where $\Sigma_0, \, \Sigma_1$ are vertical complex lines equipped with the coordinates $z$, satisfies $$\label{contr}
|({\rm Hol}(c))^{\prime}| = e^{- Re(\int_c \omega_1)} < 1 \, .$$ Clearly this formula means that the holonomy is contractive. Roughly speaking, the role played by these trajectories in our discussion can be summarized as follows. Nearby a sink $p$ all real trajectories converge to $p$. Estimate (\[contr\]) guarantees that the distance of the leaves of ${{\widetilde{\mathcal F}}}_S$ to $L_{\infty}$ has a local minimum at $p$ (which may well be zero). On the other hand, nearby a source $p$, all real trajectories go away from $p$. This means that the distance of the leaves of ${{\widetilde{\mathcal F}}}_S$ to $L_{\infty}$ reaches a local maximum at $p$. It remains to consider the case of saddle singularities of $\mathcal{H}$. Indeed it can happen that one of these trajectories meets, as a separatrix, a saddle singularity of $\mathcal{H}$. In this case the trajectory in question can be continued as follows: if our trajectory defines a separatrix for a saddle singularity of $\mathcal{H}$, this saddle singularity also possesses repelling proper directions (in number correspondent to the number of attractive ones). Thus the mentioned trajectory can naturally be continued by following a separatrix associated to a repelling direction. The continued trajectory still yields contractive holonomy. Alternately we may consider a slight perturbation of the trajectory in question on a neighborhood of the saddle singularity so as to avoid it to reach the singularity itself.

\[rem\] [More generally, it follows from the preceding that only sinks or sources singularities of $\mathcal{H}$ (corresponding to maxima or minima for the distance function between leaves) can provide genuine [*endpoints*]{} for a trajectory of $\mathcal{H}$. Otherwise we shall allow the trajectory to be continued regardless of whether or not it passes several times over the same point. In particular a regular [*periodic trajectory of $\mathcal{H}$ necessarily has infinite length.*]{}]{}

Going back to our specific case in which $\omega_1$ is characterized by Formula (\[omega1\]), it follows that the local trajectories of ${\mathcal{H}}$ are determined as the lift in $T_{(x,y(x))}L_{\infty}$ of the vector $v$ where $v$ is such that $v.H(x,y(x)) / F(x,y(x))$ belongs to ${{\mathbb R}}_-$. Also it is to be noted that the abelian form $\omega_1$ is independent of the leaf in the same cone $S$. In fact, Equation (\[omega1\]) shows that it depends solely on $L_{\infty}$. These remarks can be summarized as follows.

1.  The trajectory of ${\mathcal{H}}$ through a point $(p_1, p_2,p_3)$ projects on the trajectory of ${\mathcal{H}}$ through the point $(p_1,p_2,0)$ which, in addition, is globally contained in the plane $\{z=0\}$.

2.  Since the absolute value of the coordinate “$z$” is always decreasing over a trajectory of ${\mathcal{H}}$, it follows that the trajectory of ${\mathcal{H}}$ through $(p_1, p_2,p_3)$ has infinite length if and only if the the trajectory of ${\mathcal{H}}$ through $(p_1, p_2,0)$ has infinite length.

The following simple lemma will also be important later on. For this lemma we should take into account that, whereas $\Delta_{\infty}^{(x,y)}$ can be chosen “generic”, it always possesses points of tangency with the foliation ${{\widetilde{\mathcal F}}}_{\infty}$.

\[remainingcompact\] For a generic choice of the affine coordinates $(x,y)$ an oriented trajectory of ${\mathcal{H}}$ never intersect $\Delta_{\infty}^{(x,y)}$. Besides there is a compact part $K \subset {{\mathbb C}}^3$ and a constant $C_K$ so that the following holds: every segment of trajectory $l$ of ${\mathcal{H}}$ whose total length is greater than $C_K$ verifies the condition that the part of $l$ lying in ${{\mathbb C}}^3 \setminus K$ is less than, say, $1/10$ of the total length of the segment in question.

Let $q_1, \ldots , q_r$ be the points where $\Delta_{\infty}^{(x,y)}$ is tangent to ${{\widetilde{\mathcal F}}}_{\infty}$ and fix a small neighborhood $W_i$ of $q_i$, $i=1, \ldots ,r$. Then there is a “tubular neighborhood” $V$ of $\Delta_{\infty}^{(x,y)} \setminus \bigcup_{i=1}^r W_i$ so that the following holds: for every point $p \in \partial V \setminus \bigcup_{i=1}^r W_i$ the trajectory of ${\mathcal{H}}$ through $p$ is transverse to $\partial V$ and oriented outwards $V$. In other words, no trajectory of ${\mathcal{H}}$ may enter $V$ without entering first some $W_i$.

On the other hand the structure of ${\mathcal{H}}$ trajectories on $W_i$ is easy to describe. If $\Delta_{\infty}^{(x,y)}$ is “sufficiently generic”, then the tangency of $\Delta_{\infty}^{(x,y)}$ at ${{\widetilde{\mathcal F}}}_{\infty}$ at $q_i$ is quadratic (for all $i \in \{1, \ldots, r\}$). In particular if $L_i$ is the local leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ through $q_i$, the point $q_i$ is itself a source-type singularity for the trajectories of ${\mathcal{H}}$. Thus no trajectory of ${\mathcal{H}}$ actually intersects $\Delta_{\infty}^{(x,y)}$. Finally if $V$ and the neighborhoods $W_i$ are sufficiently small, then the length of a segment of trajectory lying in $V\cup \bigcup_{i=1}^r W_i$ is less than, say, $1/30$ the length of the segment of same trajectory in $K={{\mathbb C}}^3 \setminus V\cup \bigcup_{i=1}^r W_i$ which is determined by two “successive” passages of the trajectory in question through $V\cup \bigcup_{i=1}^r W_i$. The statement then follows.

\[lastsection3\] [In certain cases it may be useful to make a “non-generic” choice of the affine coordinates $(x,y)$ so as to have a line at infinity $\Delta_{\infty}^{(x,y)}$ passing through certain singular points of ${{\widetilde{\mathcal F}}}_{\infty}$. We shall briefly mention one situation of this type later on, cf. Remark (\[tobeincluded\]).]{}

Renormalization in the exceptional divisor
==========================================

Let us continue our discussion of homogeneous polynomial vector fields. We shall keep the notations of Section 3 emphasizing the $3$-dimensional case though all the results presented below are valid in arbitrary dimensions.

Again $X$ will stand for a homogeneous polynomial vector field of degree $d \geq 2$ whose associated foliation is denoted by ${{\mathcal F}}$. The presence of a vector field enables us to endow every regular leaf $L$ of ${{\mathcal F}}$ with an Abelian $1$-form, denoted by $dT$ and defined by the equation $dT.X =1$. The $1$-form $dT$ is going to be called the time-form induced by $X$. When $X$ is complete, given a curve $c:[0,1] \rightarrow L$ joining two points in $L$, $c(0)$ and $c(1)$, the integral $\int_c dT$ measures the time needed to cover $c$ from $c(0)$ to $c(1)$ following the flow of $X$. Again ${{\widetilde{X}}}$ stands for the vector field induced by $X$ on $M$. Throughout this section generic affine coordinates $(x,y)$ are chosen (as in the context of the previous section).

Since the vector field ${{\widetilde{X}}}$ has poles over $\Delta_{\infty}$, the time-form is not defined for the regular leaves of ${{\widetilde{\mathcal F}}}_{\infty}$. However, it is possible to define a “renormalized time-form” on a neighborhood of each regular point $p$ of a leaf $L_{\infty} \subseteq
\Delta_{\infty}$. This goes as follows. Let $L_{\infty} \subseteq \Delta_{\infty}$ be a regular leaf of ${{\widetilde{\mathcal F}}}$ and let $p \in L_{\infty}$ be a regular point of $L_{\infty}$ which is not singular for ${{\widetilde{\mathcal F}}}$. Choose local coordinates $(x',y',z')$ around $p$ where the foliation becomes locally given by the vector field $\partial / \partial x'$. Besides we also impose $\{ z'=0 \} \subset
\Delta_{\infty}$. In these coordinates the vector field ${{\widetilde{X}}}$ is given by ${z'}^{1-d} f(x',y',z')
\partial/\partial x'$. The “renormalized time-form” over $L_{\infty}$ is then defined as $dx'/f(x',0,0)$. In other words, it is obtained from ${{\widetilde{X}}}$ by eliminating its pole component. Naturally there is no canonical choice for the coordinate $z'$ and this prevents us from having a global definition for the “renormalized time-form”. More precisely the local form $dx'/f(x',0,0)$ is not globally defined on $L_{\infty}$ because, when a change of coordinates is performed, two local definitions of this “renormalized time-form” will agree only up to a multiplicative constant. Therefore, whereas the previous construction does not define an Abelian form on $L_{\infty}$, it endows $L_{\infty}$ with an affine structure (for further details we refer to [@guillotreb]). The purpose of this section is to exploit this affine structure to estimate the domain of definition of the solutions of ${{\widetilde{X}}}$. As it will be seen, precise estimates can be obtained in this way as long as the “evolution” of the coordinate “$z$” is well-controlled.

Although we have defined the “renormalized time-form” only at regular points of ${{\widetilde{\mathcal F}}}$, this form admits a natural asymptotic extension to the singularities of ${{\widetilde{\mathcal F}}}$ lying in $\Delta_{\infty}$. Details on these extensions will be given as they become necessary.

Fix a point $p_0$ contained in the singular set of ${{\widetilde{\mathcal F}}}_{\infty}$. Suppose that the restriction of ${{\widetilde{\mathcal F}}}$ to a neighborhood of $p_0$ is given by Equation (\[tXX\]) so that $${{\widetilde{X}}}= \frac{1}{z^{d-1}} \left[ F(x,y) \frac{\partial }{\partial x}
+ G(x,y) \frac{\partial }{\partial y} + zH(x,y) \frac{\partial
}{\partial z} \right].$$ With the notations of Section \[sechom\], let $\textsc{P} = {\rm
g.c.d.}\, (F,G)$ so that $F = \textsc{P}. a (x,y)$ and $G =
\textsc{P} . b (x,y)$. Denoting by $\overline{\textsc{P}}$ the greatest common divisor between $\textsc{P}$ and $H$. Thus we can set $\textsc{P} = \overline{\textsc{P}} \textsc{P}^{\ast}$ and $H = \overline{\textsc{P}} H^{\ast}$. It follows that $$\label{campo}
{{\widetilde{X}}}= \frac{\overline{\textsc{P}}}{z^{d-1}} \left[ \textsc{P}^{\ast}(x,y) \left( a (x,y)
\frac{\partial }{\partial x} + b (x,y) \frac{\partial }{\partial y}
\right) + zH^{\ast} (x , y) \frac{\partial}{\partial z} \right]$$ being $p_0 \simeq (0,0,0)$. If $\textsc{P}^{\ast}$ is not constant, the curve in $\Delta_{\infty}$ induced by $\{ \textsc{P}^{\ast} =0\}$ is singular for ${{\widetilde{\mathcal F}}}$, though its generic points are regular for ${{\widetilde{\mathcal F}}}_{\infty}$. From this point of view $\{ \textsc{P}^{\ast} =0\}$ may or may not be invariant by ${{\widetilde{\mathcal F}}}_{\infty}$. Again to abridge the discussion we shall make the following assumption valid for the rest of this paper unless otherwise stated:

[**General assumption**]{}: No irreducible component of the curve $\{ \textsc{P}^{\ast} =0\}$ is invariant by ${{\widetilde{\mathcal F}}}_{\infty}$.

This condition does not affect the general character of the results stated in the Introduction. For example, concerning the statements of Theorems A, A’ and B, we have the following:

\[aproposition\] Assume that $X$ is a homogeneous semi-complete vector field with degree greater than or equal to $3$. Suppose that ${{\widetilde{X}}}$ is as in Equation (\[campo\]). Then $\textsc{P}^{\ast} = 0$ is not invariant by ${{\widetilde{\mathcal F}}}_{\infty}$. In fact, a regular leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ cannot be contained in the singular set of ${{\widetilde{\mathcal F}}}$.

Let $L$ be a regular leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ and assume for a contradiction that $L$ is contained in the singular set of ${{\widetilde{\mathcal F}}}$. Since ${\rm g.c.d.}\, (a, b)$ is locally invertible (cf. Equation (\[campo\])), $\textsc{P}^{\ast}$ must vanish identically over $L$. Denote by $\textsc{P}_1$ the irreducible component of $\textsc{P}^{\ast}$ vanishing over $L$.

Let $m \geq 1$ be the order of $\textsc{P}^{\ast}_1$ over $L$. In other words, at a generic smooth point of $L$ we can find local coordinates $(x,y,z)$, $\{ z=0\} \subset \Delta_{\infty}$, where $L$ is identified with $\{y=0, z=0\}$. In these coordinates we have that $\textsc{P}^{\ast}_1(x,y) = y^m$ and $H(0,0) \neq 0$. Besides if the point is sufficiently generic we also have $a(0,0) \neq 0$.

Because $L$ is invariant for the foliation, it follows that $b$ is divisible by $y$. Recalling also that $m \geq 1$, it follows that, in the above local coordinates, the first non-trivial homogeneous component of ${{\widetilde{X}}}$ at the origin is given by $${{\widetilde{X}}}^H = z^{1-d} y^k \left[ {\alpha}y \frac{\partial }{\partial x} + {\lambda}z \frac{\partial }{\partial z} \right]$$ for some constants ${\lambda}= H(0,0) \in {{\mathbb C}}^{\ast}$, $k \geq 0$ and ${\alpha}\in {{\mathbb C}}$ (${\alpha}\ne 0$ if and only if $m =1$). The hyperplanes $\{y
= {\rm cte}\}$ are invariant for the associated foliation. For each nonvanishing constant ${\rm cte}$ sufficiently close to zero, the leaves are such that $$\dot{z} = {\rm cte}^k {\lambda}z^{2-d}$$ However, since $d \geq 3$, the corresponding vector field is clearly not semi-complete what contradicts the fact that $L$ is invariant by ${{\widetilde{\mathcal F}}}_{\infty}$. The proposition follows.

The above proof also yields:

\[trivializingP\] Assume that $d \geq 3$. Then $\textsc{P}^{\ast}$ must be constant provided that ${{\widetilde{X}}}$ is semi-complete. If $d=2$ and $\textsc{P}^{\ast}$ is not constant, then the foliation ${{\widetilde{\mathcal F}}}_{\infty}$ is induced by a vector field of degree $0$ or $1$.

By construction, the renormalized time-form is defined only for regular leaves of the foliation (no matter whether or not they are contained in the zero/pole divisor). The interest of the above lemma lies precisely in the statement that, under the conditions above, [*no leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ is contained in the singular set of ${{\widetilde{\mathcal F}}}$*]{} so that the “renormalized time-form” can be defined over every leaf $L_{\infty} \subseteq \Delta_{\infty}$. Furthermore if $X$ is semi-complete but has degree $2$, the fact that $\textsc{P}$ is not invertible implies that $X$ is a linear function times a linear vector field. Clearly this case can directly be treated so that it is going to be left to the reader. Summarizing for the rest of this section and for all of Section 5, the “general assumption” stated above will be assumed without further comments.

Consider the foliation ${{\widetilde{\mathcal F}}}_{\infty}$ on $\Delta_{\infty}$. Fix a regular leaf $L_{\infty} \subseteq \Delta_{\infty}$ of ${{\widetilde{\mathcal F}}}$ and let $S$ be the cone over $L_{\infty}$, $\mathcal{P}^{-1}_{\infty}(L_{\infty})$. Denote by $\mathcal{H}$ the oriented $1$-dimensional real foliation induced by the Abelian form $\omega_1$ (cf. Section \[sechom\]). It is also useful to consider other foliations similar to $\mathcal{H}$. For this let us consider an angle $\theta \in (-\pi/2, \pi/2)$. Denote by $\mathcal{H}^{\theta}$ the oriented foliation whose (oriented) trajectories make an angle of $\theta$ with the (oriented) trajectories of $\mathcal{H}$. It is clear that these foliations are well-defined under the same conditions that $\mathcal{H}$. It is also clear that the holonomy maps of ${{\widetilde{\mathcal F}}}_S$ obtained over the trajectories of $\mathcal{H}^{\theta}$ are still contractions as in Formula (\[contr\]) (up to a constant). In the sequel we denote by $l^{\theta}$ an oriented trajectory of $\mathcal{H}^{\theta}$.

Given (a segment of) a trajectory $l_p$ of $\mathcal{H}$ (resp. $l_p^{\theta}$ of $\mathcal{H}^{\theta}$), we are interested in the value of the integral $\int_{l_p} dT$ (resp. $\int_{l_p^{\theta}} dT$). To investigate the behavior of this integral, it is clear that the singularities of ${{\widetilde{\mathcal F}}}_{\infty}$ will pose further difficulties. Thus it is natural to begin with (segments of) trajectories of $\mathcal{H}$ (resp. $\mathcal{H}^{\theta}$) that remain away from the corresponding singular set. For this let $W$ be a sufficiently small open neighborhood of the singular set of ${{\widetilde{\mathcal F}}}$ on $\Delta_{\infty}$. Let $l_p$ (resp. $l_p^{\theta}$) be (a segment of) a trajectory of ${\mathcal{H}}$ (resp. ${\mathcal{H}}^{\theta}$). We can now state one of our main results. Despite our $3$-dimensional setting, the reader will immediately check that this result holds in arbitrary dimensions (as it is always the case in the present section).

\[introducelabel4\] Suppose that $l_p$ (resp. $l_p^{\theta}$) is contained in $\Delta_{\infty} \setminus W$. Then $\int_{l_q} dT$ (resp. $\int_{l_q^{\theta}} dT$) converges for all $q = (p,z_0) \in \mathcal{P}^{-1}_{\infty}(p,0)$, where $l_q$ (resp. $l_q^{\theta}$) denotes the lift of $l_p$ (resp. $l_p^{\theta}$) to the leaf of ${{\widetilde{\mathcal F}}}$ through $q$ and where $dT$ stands for the time form associated to ${{\widetilde{X}}}$. More precisely, assuming $W$ fixed, there exists a constant $C$ (varying continuously with $\theta$) such that for every path $c :[0,1] \rightarrow L$, $c(0) =q$, with image contained in a trajectory $l_q^{\theta}$ of ${\mathcal{H}}^{\theta}$ and not intersecting $W$, we have $$\int_c dT \leq \int_{l_q^{\theta}} dT < C \vert z_0 \vert^{d-1} \, .$$

It suffices to prove the statement for the case of a trajectory $l_p$ of ${\mathcal{H}}$ since the adaptations needed for trajectories of ${\mathcal{H}}^{\theta}$ are clear. Also we can suppose without loss of generality that the length of $l_p$ is infinite. Finally we recall that the affine coordinates $(x,y)$ were generically chosen in the sense of Section 3. Also let $W$ be the previously chosen open neighborhood of the singular set of ${{\widetilde{\mathcal F}}}$ on $\Delta_{\infty}$. Assume that $l_p$ is connected and totally contained in $\Delta_{\infty} \setminus W$. Since the intersection of $\Delta_{\infty} \setminus W$ with the singular set of ${{\widetilde{\mathcal F}}}_{\infty}$ is empty, the only singularities of $\mathcal{H}$ that can appear on $l_p$ are saddles. However, as noted in Remark \[rem\], the corresponding trajectories of $\mathcal{H}$ can locally be deformed to avoid these singularities. Moreover the uniform contractive character of the corresponding holonomy maps is still kept by these deformed trajectories. As to the polar set of $\omega_1$, we recall that this set is contained in the line at infinity $\Delta_{\infty}^{(x,y)}$. Thanks to Lemma \[remainingcompact\], every sufficiently long segment of $l_p$ remains most “of its length” on a fixed compact part of the affine ${{\mathbb C}}^2$ associated to the coordinates $(x,y)$. Since $F$ is clearly bounded on this compact part, the estimates of Lemma \[remainingcompact\] allow us to conclude the following: every sufficiently long segment $c_p$ of $l_p$ can be split into a concatenation $c_1 \ast c_2 \ast \cdots \ast c_k$ such that:

1.  The image of $c_i$, for $i$ odd, is contained in the compact set $K$. Besides at points belonging to these segments the absolute value of $\omega_1$ is bounded from below, i.e. $|\omega_1| \geq {\alpha}> 0$.

2.  If $i_0$ is odd, then the sum of the lengths of all even $i$’s, $i <i_0$, is less than, say, $2/3$ the sum of the lengths of $c_1, \ldots ,c_{i_0}$.

3.  The absolute value of the coordinate “$z$” decreases monotonically over the segment $c_p$.

Fix $q \in \mathcal{P}^{-1}_{\infty}(p)$ and let $L$ be the leaf through $q$. Consider the lift of $l_p$ to $L$ and denote it by $l_q$. Note that $l_q$ is an oriented trajectory of $\mathcal{H}$ over $L$. We want to express $l_q$ in the coordinates $(x,y,z)$ considered in the preceding section. In fact, the goal will be to compute the value of its last coordinate $z$. For this consider a connected oriented path $c$ contained in $l_p$, joining $p$ to another point of $l_p$. Consider also a lift of $c$ contained in $l_q$. The $z$-coordinate of the mentioned lift is given by $$z = z_0 e^{-\int_c \omega_1}$$ where $z_0$ is the $z$-coordinate of $q$. In other words, $z_0$ is the “height" of $q$ relatively to $L_{\infty}$. In particular $$\begin{aligned}
|z| &=& \left|z_0 e^{-\int_c \omega_1}\right| = |z_0| e^{-{\rm Re}
\int_c \omega_1} = |z_0| e^{-\int_0^1 {\rm Re} (\omega_1(c(t)).
c^{\prime}(t)) dt}\\
&=& |z_0| e^{-\int_0^1 |(\omega_1(c(t)). c^{\prime}(t)| dt} \leq
|z_0| e^{-\int_0^1 {\alpha}|c^{\prime}(t)| dt /3} = |z_0| e^{-{\alpha}{\rm
length}(c) /3}\end{aligned}$$ This estimate shows that, whenever a segment of $l_p$ having length equal to $3\ln(2)/ 2{\alpha}$ is lifted in a regular leaf of ${{\widetilde{\mathcal F}}}$ projecting over $L_{\infty}$, the height of the final point of the lift in question is at most $1/2$ of the height of its initial point.

Now the integral $\int_{l_q} dT$ can be estimated as follows. The time-form on $L$ is given, in local coordinates, by $$dT = \frac{z^{d-1}}{F(x,y)}dx.$$ Since $l_p$, the projection of $l_q$ by $\mathcal{P}_{\infty}$, is contained on a compact set not intersecting the singular set of ${{\widetilde{\mathcal F}}}_{\infty}$, the absolute value of $F(x,y)$ is bounded from below, i.e. $$|F(x,y)| \geq {\beta}> 0, \qquad {\rm for \;\; \; all } \; \; \; (x, y)
\in \Delta_{\infty} \setminus W.$$ Otherwise we replace $F$ by $G$ (recall that we are dealing only with regular points of ${{\widetilde{\mathcal F}}}_{\infty}$). Hence, considering $l_q$ as the concatenation of segments having length equal to $3\ln(2)
/2 {\alpha}$, $l_q = \sum_{i=0}^{\infty}l_{i,q}$, it follows that $$\begin{aligned}
\left| \int_{l_q} dT \right| &=& \left| \sum_{i=0}^{\infty} \int_{l_{i,q}} \frac{z^{k-1}}{F(x, y)} dx \right|
\leq  \sum_{i=0}^{\infty} \left| \int_0^1 \frac{z_{i,q}^{d-1}(t)}{F(x_{i,q}(t),y_{i,q}(t))} x^{\prime}_{i,q}(t) dt \right|\\
&\leq&  \sum_{i=0}^{\infty} \int_0^1 \frac{|z_{i,q}(t)|^{d-1}}{|F(x_{i,q}(t),y_{i,q}(t))|} |x^{\prime}_{i,q}(t)| dt
\leq  \sum_{i=0}^{\infty} \int_0^1 \frac{|z_0|^{d-1}(\frac{1}{2})^{i(d-1)}}{{\beta}} |l^{\prime}_{i,p}(t)| dt\\
&\leq&  \frac{|z_0|^{d-1}}{{\beta}} {\rm length }(l_{i,p}) \sum_{i=0}^{\infty}\left( \frac{1}{2^{d-1}} \right)^{i}
=  \frac{3|z_0|^{d-1} {\rm ln}(2)}{2{\alpha}{\beta}} \frac{1}{1 - (\frac{1}{2})^{d-1}} < \infty\end{aligned}$$ where $l_{i,q}(t) = (x_{i,q}(t), y_{i,q}(t), z_{i,q}(t))$, $t \in
[0,1]$, is such that $l_q = \sum_{i=0}^{\infty} l_{i,q}$ and $\mathcal{P}_{\infty}(l_{i,q}) = l_{i,p}$. The proposition follows.

What precedes shows that the integral mentioned above is indeed bounded on $\Delta_{\infty} \setminus W$. Our next goal will be to get rid of the condition on $W$, i.e. we want to allow the trajectory $l_p$ (resp. $l_p^{\theta}$) to accumulate on the singular set of ${{\widetilde{\mathcal F}}}$ in $\Delta_{\infty}$. This will lead us to study the behavior of this integral over segments of trajectories of ${\mathcal{H}}$ (resp. ${\mathcal{H}}^{\theta}$) that are close to the singularities of ${{\widetilde{\mathcal F}}}$. This local analysis will be the object of the Section 4. Nonetheless, to finish the current section, let us give an elementary general result concerning trajectories ${\mathcal{H}}, \, {\mathcal{H}}^{\theta}$ that are contained in a local separatrix for a singularity of ${{\widetilde{\mathcal F}}}, \, {{\widetilde{\mathcal F}}}_{\infty}$. This goes as follows. Consider again a vector field ${{\widetilde{X}}}$ as in Equation (\[campo\]). Let $p \in \Delta_{\infty}$ be a singular point of ${{\widetilde{\mathcal F}}}$ and consider a (germ of) analytic curve $Sep \subset \Delta_{\infty}$ passing through $p$ and not entirely contained in the singular set of ${{\widetilde{\mathcal F}}}$. Besides let $\gamma (t)$ denote an irreducible Puiseaux parametrization for $Sep$ defined on a neighborhood of $0 \in {{\mathbb C}}$. Finally denote by $k$ the order at $0 \in {{\mathbb C}}$ of the vector field obtained by pulling-back the restriction of the vector field $\textsc{P}(x,y) ( a (x,y) \frac{\partial }{\partial x} + b (x,y) \frac{\partial }{\partial y} )$ to $Sep$ by $\gamma$. Our assumption concerning $Sep$ ensures that $k$ is a strictly positive integer. Now we have:

\[separatrixndimensions\] Consider the restriction of $\omega_1$ to $Sep$. Then the nature of $\omega_1$ (restricted to $Sep$) at $p$ is determined by the order $l$ at $0 \in {{\mathbb C}}$ of the function $t \mapsto H \circ \gamma (t)$ as follows:

-   If $l =k$ then $\omega_1$ is holomorphic and regular at $p$. If $l > k$ then $\omega_1$ is holomorphic and the restriction of ${\mathcal{H}}$ to $Sep$ has a saddle singularity at $p$.

-   If $l=k-1$ then $\omega_1$ has a simple pole at $p$. The residue of the pole is $-H(0,0) /F(0,0)$. In particular the restriction of ${\mathcal{H}}$ to $Sep$ has a sink (source) at $p$ provided that $H(0,0)/F(0,0) \in {{\mathbb R}}_+$ (resp. $H(0,0)/F(0,0) \in {{\mathbb R}}_+$).

-   If, in addition, $X$ is semi-complete and $l < k-1$ then $d=2$. Furthermore $\omega_1$ has a pole of order 2 or greater at $p$. The foliation ${\mathcal{H}}$ has a saddle singularity at $p$.

The cone over $Sep$ can locally be parameterized by $(t, z) \mapsto (\gamma (t) ,z)$. Since this cone is left invariant by ${{\widetilde{\mathcal F}}}$, we can pull-back (the restriction of) ${{\widetilde{\mathcal F}}}, \, {{\widetilde{X}}}$ to the coordinates $(t,z)$. We then obtain a vector field $Y$ given by $$Y = z^{1-d} [ f(t) \partial /\partial t + z(H \circ \gamma) (t) \partial /\partial z]$$ where the order at $0 \in {{\mathbb C}}$ of the function $f(t)$ is $k$. In these coordinates the form $\omega_1$ restricted to the axis $\{ z=0\}$ is nothing but $H \circ \gamma (t) dt/f(t)$. Therefore to conclude the statement it suffices to check that $d=2$ for $l < k-1$. To do this, note that the condition $k > l+1$ implies that the first non-trivial homogeneous component of ${{\widetilde{X}}}$ at $p$ has the form $z^{2-d} H \circ \gamma (t) \partial /\partial z$. However the latter vector field is not semi-complete for $d \geq 3$.

To close this section let us point out again that the preceding statements hold in arbitrary dimensions despite our choice of emphasizing the case of dimension $3$. The reader will have no difficulty to realize that our proofs work equally well in the $n$-dimensional case.

Structure of $\mathcal{H}$ near singular points of ${{\widetilde{\mathcal F}}}_{\infty}$
========================================================================================

Next we are going to establish an extension of Theorem \[introducelabel4\] by allowing the trajectories of ${\mathcal{H}}, \, {\mathcal{H}}^{\theta}$ to accumulate on singularities of ${{\widetilde{\mathcal F}}}, \, {{\widetilde{\mathcal F}}}_{\infty}$. These singularities however will be supposed to be “simple enough”. These assumptions are, in fact, necessary since reasonably simple singularities, such as “saddle-nodes” slightly more degenerated than those considered in the statement of Theorem B already give rise to new complications preventing us from generalizing Theorem \[introducelabel4\] without further information.

As in Sections 3 and 4, we still keep notations and provide arguments emphasizing the $3$-dimensional case. The extensions of the arguments to higher dimensions however does not present any additional difficulty. Again we consider the foliation ${{\widetilde{\mathcal F}}}$ associated to a homogeneous semi-complete vector field on ${{\mathbb C}}^n$ and assume that the singularities of ${{\widetilde{\mathcal F}}}$ lying in $\Delta_{\infty}$ are simple in the sense stated in the Introduction. In particular, if $n=3$, then this condition is equivalent to saying that the singularities of ${{\widetilde{\mathcal F}}}_{\infty}$ possess exactly two eigenvalues different from [*zero*]{}. Moreover if these eigenvalues have the form $1,N$ with $N \in \mathbb{Z}_+$, then we also ask the singularity to be linearizable (i.e. not to be conjugate to its Poincaré-Dulac normal form, cf. for example [@arnold]). With these assumptions Theorem \[introducelabel4\] admits the following extension:

\[maintheo\] Suppose that $X$ (homogenous) is semi-complete and assume that all the singularities of ${{\widetilde{\mathcal F}}}_{\infty}$ are simple (in the sense indicated in the Introduction). Suppose that there is $\theta \in (-\pi/2, \pi/2)$ and a point $p\in \Delta_{\infty}$ such that the trajectory $l_p^{\theta}$ of $\mathcal{H}^{\theta}$ through $p$ has infinite length. Then $\int_{l_q} dT$ converges for all $q \in \mathcal{P}^{-1}_{\infty}(p)$, where $l_q$ denotes the lift of $l_p$ to the leaf through $q$ and $dT$ is the time form associated to ${{\widetilde{X}}}$.

Applications of Theorems \[introducelabel4\] and \[maintheo\] will be provided in the forthcoming sections. The present section will entirely be devoted to the proof of Theorem \[maintheo\].

In view of Proposition \[aproposition\] (cf. also Corollary \[trivializingP\]), we can assume that $\textsc{P}^{\ast}$ is constant (and thus equal to $1$) so that $\textsc{P} = \overline{\textsc{P}}$. Hence $X$ is a homogeneous polynomial vector field multiplied by a homogeneous polynomial. Since multiplying a homogeneous vector field by a homogeneous polynomial does not affect our arguments, which are essentially determined by the underlying foliation, we are going to assume throughout the sequel that $\textsc{P} = \overline{\textsc{P}}\textsc{P}^{\ast} \equiv 1$ to abridge notations.

To begin with fix a point $p_0 \in \Delta_{\infty}$ contained in the singular set of ${{\widetilde{\mathcal F}}}$. Recall that the two eigenvalues of ${{\widetilde{\mathcal F}}}_{\infty}$ at $p_0$ are supposed to be different from zero (and, in case they are of the form $1,N$ with $N \in \mathbb{Z}_+$, ${{\widetilde{\mathcal F}}}_{\infty}$ is supposed not to be conjugate to its Poincaré-Dulac normal form). To be more precise, a vector field representing ${{\widetilde{\mathcal F}}}$ about $p_0$ and having a singular set of codimension $\geq 2$ can be written as $$\bar{X} =  F (x,y) \frac{\partial }{\partial x} + G (x,y)
\frac{\partial }{\partial y} + zH(x,y) \frac{\partial }{\partial z}$$ with ${\rm g.c.d.}\, (F,G) = 1$. Note that $\bar{X}$ differs from our vector field ${{\widetilde{X}}}$ by a multiplicative function having the form $z^{1-d} f(x,y)$ where $f(0,0) \neq 0$. The condition about the singularities of ${{\widetilde{\mathcal F}}}_{\infty}$ simply means that the vector field $Z = F (x,y) \partial /\partial x + G (x,y) \partial
/\partial y$ has eigenvalues ${\lambda}_1, \, {\lambda}_2$ at $p_0$ with ${\lambda}_1 {\lambda}_2 \neq 0$. Furthermore if ${\lambda}_1, \, {\lambda}_2$ is of the form $1,N$ with $N \in \mathbb{Z}_+$, then $Z$ is linearizable (recalling that a non-linearizable vector field verifying the preceding conditions must be conjugate to $(Nx + y^N) \partial /\partial x
+ y \partial /\partial y$). This summarizes the assumption of Theorem \[maintheo\].

Fixed a separatrix $Sep$ for ${{\widetilde{\mathcal F}}}_{\infty}$ at $p_0 \in \Delta_{\infty}$, we have:

\[thislabelwasmissing\] Assume that $H(0,0) = 0$. Then the Abelian form $\omega_1$ on the cone over $Sep$ is holomorphic.

Since $p_0$ has non-degenerate linear part for ${{\widetilde{\mathcal F}}}_{\infty}$, it follows that whether or not $Sep$ is smooth, the order $k$ at $0 \in {{\mathbb C}}$ of the vector field obtained by pulling-back the restriction of the vector field $Z$ to $Sep$ by an irreducible Puiseaux parameterization $\gamma$ equals $1$. The statement then results from Lemma \[separatrixndimensions\].

It follows from this lemma that $p_0$ is either a regular point or a saddle singularity for ${\mathcal{H}}$. In both cases the singular point in question can be avoided without disrupting the contractive character of the corresponding holonomy map (Remark \[rem\]). In particular this type of singularity can be ruled out from the subsequent discussion. Let us now consider the case where $H(0,0) \ne 0$.

\[hreal\] For $X$ semi-complete assume that $H(0,0) \neq 0$. Then $d=2$ and $H(0,0)/{\lambda}_1 \in {{\mathbb Q}}$.

Let us consider the restriction of ${{\widetilde{X}}}$ to the invariant manifold $\{x=0, \, y=0\}$. This restriction is naturally a semi-complete vector field that does not vanish identically, since we are assuming $H(0,0) \ne 0$. The restriction is, in fact, given by $H(0,0) z^{2-d} {\partial /\partial z}$. The semi-complete assumption implies that $d=2$.

Let us now denote by ${{\widetilde{\mathcal F}}}_S$ the foliation induced on the $2$-plane sitting over the invariant manifold $\{y=0\}$. Obviously $Sep$ is a separatrix for ${{\widetilde{\mathcal F}}}_S$. The restriction of ${{\widetilde{X}}}$ to this plane is clearly semi-complete. Besides this restriction is not identically zero since ${{\widetilde{X}}}= z^{1-d} \bar{X}$ up to an invertible factor. In particular the first homogeneous component at the origin of this restriction is therefore semi-complete as well. Nonetheless, the homogeneous component in question is given by $z^{-1} [ {\lambda}_1 x {\partial /\partial x}+ H(0,0) z {\partial /\partial z}]$. Now note that its asymptotic order, which is equal to $1 - H(0,0)/{\lambda}_1$, must belong to $[0,2] \cap {{\mathbb Q}}$ since the vector field is semi-complete. The result follows.

In particular the quotient of the eigenvalues of ${{\widetilde{\mathcal F}}}_{\infty}$ at the singular point $p_0$ is a rational number (and thus real). In this case the $1$-form $\omega_1$ has a simple pole at the origin ($\simeq p_0$). Therefore there are two cases to be considered according to whether ${\lambda}_1/{\lambda}_2 \in {{\mathbb Q}}_+$ or ${\lambda}_1/{\lambda}_2 \in {{\mathbb Q}}_-$. The first case is easy to be treated.

\[quotientinR+\] With the preceding notations suppose that ${\lambda}_1/{\lambda}_2 \in {{\mathbb Q}}^+$. Then $p_0$ is a sink $($resp. source$)$ singularity for ${\mathcal{H}}$ provided that $H(0,0)/{\lambda}_1 > 0$ $($resp. $H(0,0)/{\lambda}_1 < 0)$. In either case, $p_0$ yields a terminal point for the trajectories of ${\mathcal{H}}$.

It suffices to consider the case $H(0,0)/{\lambda}_1 > 0$. Clearly the structure of ${\mathcal{H}}$ over the two (smooth) separatrices of ${{\widetilde{\mathcal F}}}_{\infty}$ at $p_0$ corresponds to sinks. As to the remaining leaves, recall that they all accumulate on the origin. Furthermore the structure of ${\mathcal{H}}$ over regular points of these leaves has to be of the same nature as the corresponding structure over the smooth separatrizes. That is to say that all these trajectories point inward the singularity $p_0 \simeq (0,0)$. The lemma is proved.

So let us assume from now on that ${\lambda}_1/{\lambda}_2 \in {{\mathbb Q}}^-$. The restriction of ${{\widetilde{\mathcal F}}}_{\infty}$ to a neighborhood of $p_0$ admits $2$ separatrices. These separatrices are the unique leaves (of the restriction of ${{\widetilde{\mathcal F}}}_{\infty}$ to a neighborhood of $p_0$) “radially” accumulating on the singular point $p_0$. In vague terms, the separatrices are the only leaves of ${{\widetilde{\mathcal F}}}_{\infty}$ accumulating on $p_0$ if we ignore the effect of the local holonomy of this foliation. Denote by $Sep$ one of them. The restriction of $\mathcal{H}$ to $Sep$ may have a singular point at $p_0 \in Sep$. The nature of this singular point depends also on the sign of the quotient $H(0,0)/{\lambda}_1$. If $H(0,0)/{\lambda}_1 > 0$ then $p_0$ corresponds to a sink of $\mathcal{H}$ (or of $\omega_1$ by a small abuse of notation) over $Sep$. Conversely, in the case where $H(0,0)/{\lambda}_1 < 0$, the singular point corresponds to a source. We note however that $H(0,0)/{\lambda}_1$ and $H(0,0)/{\lambda}_2$ have opposite signs. This implies that if $p_0$ is a sink of $\omega_1$ for one of the separatrices then $p_0$ is a source for the other.

The above indicated issue about source and sinks singularities appearing on the two separatrices of a singularity $p_0$ as before deserves some comments. First, if we consider real trajectories of $\mathcal{H}$ in the separatrix admitting $p_0$ as a sink, then these trajectories will reach a future endpoint at $p_0$. Somehow compensating the existence of this future endpoint, in the other separatrix new ${\mathcal{H}}$-trajectories are issued. These phenomena can however occur for only finitely many leaves of our foliation since each separatrix of a singularity as above can give rise to only one global leaf of ${{\widetilde{\mathcal F}}}, \, {{\widetilde{\mathcal F}}}_{\infty}$. In particular it will play no significant role in the proof of any of the theorems stated in the Introduction. In this concern, a far more important observation concern those ${\mathcal{H}}$-trajectories whose projection on $\Delta_{\infty}$ enters a small neighborhood of $p_0$ but are not contained in the corresponding separatrix of $p_0$. In fact, these trajectories can naturally be continued through the “saddle” associated to the singularity so as to eventually leave a fixed neighborhood of $p_0$. Indeed the foliation $\mathcal{H}$ is regular over all leaves of ${{\widetilde{\mathcal F}}}_{\infty}$ different from the two separatrices on a neighborhood of $p_0$. Besides, as we are going to see next, the “continued” trajectory keeps the contractive character of its holonomy.

Suppose then that the eigenvalues ${\lambda}_1, {\lambda}_2$ at $p_0$ satisfies ${\lambda}_1/{\lambda}_2 \in {{\mathbb Q}}^-$. Let us still assume that $H(0,0) \ne 0$ so that it can be normalized to be $1$. Let $U_{{\varepsilon}} = \{(x,y,z): |x|, |y| \leq {\varepsilon}\}$ be a small neighborhood of the origin $\simeq p_0$, not containing other singular points of ${{\widetilde{\mathcal F}}}_{\infty}$. Fix a regular leaf $L_{\infty} \subseteq \Delta_{\infty}$ intersecting $U$ and consider a real trajectory $l \subseteq L_{\infty}$. For these singularities we have:

\[siegeldomain\] The integral $\int_{l_q \cap U} dT$ is uniformly bounded for every $p \in l$ and $q \in {{\mathcal P}}^{-1}_{\infty}(p)$.

\[explainingstatement\] [It should be emphasized that the trajectory $l_q$ in the statement is viewed as a global trajectory of $\mathcal{H}$. In other words, the intersection $l_q \cap U$ possesses, in general, infinitely many connected components. The proposition indeed claims that the sum of the integrals of $dT$ over all these connected components is uniformly bounded.]{}

[*Proof of Proposition \[siegeldomain\]*]{}. Without loss of generality we set ${\lambda}_1 \in {{\mathbb R}}^+$. Since, locally, the only codimension $1$ component of the divisor of zeros/poles of ${{\widetilde{X}}}$ is constituted by the hyperplane at infinity (in coordinates given by $\{z=0\}$), it follows that ${{\widetilde{X}}}$ can be written as $$z^{1-d} \left[ F(x,y) \frac{\partial}{\partial x} + G(x,y)
\frac{\partial }{\partial y} + zH(x,y) \frac{\partial }{\partial z}
\right]$$ in suitable coordinates. In the above formula, we have $F(x,y) = x({\lambda}_1 + {\rm h.o.t.})$ and $G(x,y) = y({\lambda}_2 + {\rm h.o.t.})$.

Recall that ${\lambda}_1 > 0$ (resp. ${\lambda}_2 < 0$). Consider now the restriction of $\omega_1$ to the $x$-axis (resp. $y$-axis). The residue of $\omega_1$ at $0 \simeq p_0$ with respect to the mentioned axis is equal to $-H(0,0)/{\lambda}_1$ (resp. $-H(0,0)/{\lambda}_2$). Then it follows that the restriction of ${\mathcal{H}}$ to the $x$-axis (resp. $y$-axis) possesses a sink singularity (resp. source singularity) at $p_0 \simeq 0$. Hence the real trajectories contained in the $x$-axis approaches $p_0$. Similarly, those trajectories contained in the $y$-axis move away from $p_0$. It is easy to describe the behaviour of ${\mathcal{H}}$ on the regular leaves of $U$ not accumulating at $p_0$: over a real trajectory of ${\mathcal{H}}|_U$ the absolute value of $x$ decreases while the absolute value of $y$ increases. In other words, a real trajectory moves away from the invariant plane $\{y=0\}$ while approaches the plane $\{x=0\}$. In particular, whenever a (global) real trajectory $l$ enters the open set $U_{{\varepsilon}}$ it necessarily leaves $U_{{\varepsilon}}$ as well.

In this way the only possibility for a real trajectory (not contained in the global leaves containing the axes $\{ y=z=0\}$ and $\{x=z=0\}$) to accumulate on the singular point $p_0$ happens when this trajectory enters infinitely many times the open set $U_{{\varepsilon}}$. The sequence of points defined by the moment in which the mentioned trajectory enters $U_{{\varepsilon}}$ must also contain a subsequence that converges for a point in the $x$-axis. Also, in this case, it is immediate to check that the [*length of each connected component of $l \cap U_{{\varepsilon}}$*]{} is bounded above by some uniform constant.

For each leaf of ${{\widetilde{\mathcal F}}}\cap U$ not contained in the invariant plane $\{x=0\}$ the time-form is given by $$\label{timeform}
dT = \frac{z^{d-1}}{F(x,y)}dx.$$ The leaf can locally be parametrized by the $x$-variable under the form $(x, y(x), z(x))$ where $z$ is given by Equation (\[omega1\]). The form of $F$ and $G$ allows us to see that $y(x) = y_0 (x/x_0)^{{\lambda}_2/{\lambda}_1}g(x)$ for some bounded holomorphic function $g$ on ${{\mathbb C}}\setminus {{\mathbb R}}^-_0$ verifying $\lim_{x \rightarrow x_0}g(x) = 1$. In turn, the coordinate $z$ is given by $z = z_0 e^{ -\int_{x_0}^x  \omega_1 }$, where $\omega_1$ coincides with $-H(x,y(x))/F(x,y(x))dx$. Therefore, substituting $y$ and $z$ on Equation (\[timeform\]), we obtain $$dT  = z_0^{d-1} \frac{1}{F(x,y(x))} e^{ -(d-1) \int_{x_0}^x \omega_1
} dx. \label{introducelabel1}$$

Since we need to estimate the integral of the time-form over oriented real trajectories of ${\mathcal{H}}$, let us start by controlling the exponential term. Since $H(0,0) = 1$, it follows that $$-\frac{H(x,y)}{F(x,y)} = -\frac{1}{{\lambda}_1 x} \left( 1 + p(x,y)
\right)$$ for some holomorphic function $p(x,y)$ on a neighborhood of the origin verifying $p(0,0) = 0$. In particular, if $\varepsilon$ is sufficiently small, the absolute value of $p(x,y)$ is bounded above by a small constant $0 < \delta << 1$ on $U_{{\varepsilon}}$. If $l$ is a trajectory of ${\mathcal{H}}$ then $\int_l \omega_1$ is a positive real number. Therefore $$\left| e^{-(d-1)\int_l \omega_1} \right| = e^{-(d-1) {\rm Re}\int_l
\omega_1} = e^{-(d-1)\int_l \omega_1}.$$

Consider a (connected) segment of the real trajectory $l$ joining $x_0$ to $x$ where both points are contained in the neighborhood in question. Denote by $\phi : [0, 1] \rightarrow L$ a parametrization of this segment satisfying $\phi(0) = x_0$ and $\phi(1) = x$. Up to a change of coordinates, “close" to a rotation, we can assume that the (connected) segment $\phi ([0,1])$ is totally contained in the real axis. In fact, we can assume that it is contained on its positive component. In particular, we can take $\phi(t) = x_0 + t(x - x_0)$. It then follows $$\begin{aligned}
\int_l \omega_1 &= \int_0^1 \omega_1. \phi = \int_0^1 -\frac{\phi^{\prime}(t)}{{\lambda}_1 \phi(t)} (1 + p(\phi(t),
y(\phi(t))) dt\\
&= \frac{1}{{\lambda}_1} \int_0^1 -\frac{x - x_0}{x_0 + t(x - x_0)} (1 +
p(\phi(t), y(\phi(t))) dt\\
&\geq \frac{1 - \delta}{{\lambda}_1} \int_0^1 -\frac{x - x_0}{x_0 + t(x -
x_0)}dt = \frac{1 - \delta}{{\lambda}_1} \ln \left( \frac{x_0}{x} \right) \, .\end{aligned}$$ Therefore we obtain $$\left| e^{ -(d-1) \int_l \omega_1 } \right| \leq C x^{\frac{(d-1)(1
- \delta)}{{\lambda}_1}} \label{introducelabel2}$$ where $C$ is a constant depending on $d$, ${\lambda}_1$, $\delta$ and $x_0 (= {\varepsilon})$. In more accurate terms, $C = {\varepsilon}^{\frac{(1-d)(1-\delta)}{{\lambda}_1}}$. In fact, this estimation should be multiplied by a constant representing the supremum of the absolute value of the determinant of the change of coordinates. However we can, basically, include this quantity on $C$ since the absolute value of the determinant is bounded above on $U_{{\varepsilon}}$. Indeed its value is very close to $1$ since the change of coordinates is “close" to a rotation. In this sense, the constant $C$ does not depend on the segment of the real trajectory.

Now recall that $F(x,y) = {\lambda}_1 x(1 + f(x,y))$, for some holomorphic function $f$ on $U_{{\varepsilon}}$ satisfying $f(0,0)=0$. Modulo reducing ${\varepsilon}$, we can assume that $|f(x,y)|$ is bounded above by a small constant $0 < \tau << 1$. Therefore, the coefficient of the time-form satisfies $$|dT| \leq |z_0|^{d-1} \frac{C}{{\lambda}_1 (1 - \tau)} x^{\frac{(d-1)(1 -
\delta)}{{\lambda}_1} - 1} \, .$$ Since the exponent of $x$ is greater than $-1$, the primitive of the coefficient of the time-form, up to the term $z_0^{d-1}$, is a bounded holomorphic function. Consequently, the integral of the time-form, up the same term, over each connected component $l_i$ of $l
\cap U_{{\varepsilon}}$ is bounded above. In fact, there is a positive constant $K$, not depending on the trajectory of ${\mathcal{H}}$, such that $$\left| \int_{l_i} z_0^{1-d} dT \right|< K.$$

Finally the integral of the time form along $l_i$ is now bounded by $K$ times the absolute value of a positive power of the variable $z$ in the moment that the trajectory $l$ enters the open set $U_{{\varepsilon}}$ or, equivalently, on the starting point of $l_i$. We denote by $z_i$ the value $z$ at the starting point of $l_i$. As already mentioned, the holonomy of ${{\widetilde{\mathcal F}}}$ is contractive. Therefore, since the length of the real trajectory between two consecutive starting points of $l \cap U_{{\varepsilon}}$ is bounded from below, the sequence $z_i$ is such that $|z_{i+1}|/|z_i|
\leq k$, for some constant $0 < k < 1$, since the trajectories of $\mathcal{H}$ have contractive holonomy. Thus $$\left| \int_{l \cap U} dT \right| \leq \sum \left| \int_{l_i} dT
\right| \leq \sum K|z_i|^{d-1} \leq \sum K|z_0|^{d-1}k^{i(d-1)} =
\frac{K|z_0|^{d-1}}{1-k^{d-1}}$$ which means that the integral is uniformly bounded as desired.

Let us now provide the proof of Theorem \[maintheo\].

The proof follows immediately from the combination of Theorem \[introducelabel4\] with Proposition \[siegeldomain\].

[**$\bullet$ A comment about a natural relaxation of the condition on the singularities of ${{\widetilde{\mathcal F}}}$**]{}.

To close this section, and before beginning the applications of the material so far developed, we would like to indicate a far more general setting to which the previous results, especially Theorem \[maintheo\] and Proposition \[siegeldomain\], can be extended. The point of this generalization is that, to large extent, the results valid for “simple singularities” will remain valid in the context of “absolutely isolated singularities”, cf. [@canoetc].

To explain how to work out the mentioned generalizations, first consider the case where the vector field ${{\widetilde{X}}}$ has the form $$\label{tXXxy}
x^n y^m z^{1-d} \left[ F(x,y) \frac{\partial}{\partial x} + G(x,y)
\frac{\partial }{\partial y} + zH(x,y) \frac{\partial }{\partial z}
\right]$$ for some strictly positive integers $m,n$ and for $F,G$ as above. We would like to point out that, unfortunately, in this case the statement of Proposition \[siegeldomain\] is no longer valid even for (locally) semi-complete vector fields. However, the statement still holds if $m, n$ are negative integers. This is a key to adapt Theorem \[maintheo\] and Theorem B to the large class of vector fields leading to foliations whose singularities are “absolutely isolated”. In fact, consider (non-homogeneous) polynomial vector fields having only “absolutely isolated singularities” in the “hyperplane at infinity” (cf. Section 6.2), then these singularities “at infinity” can be reduced through a sequence of punctual blow-ups. In turn this will (almost always) yield a divisor over which the transform of the initial vector field will have poles. In other words, locally around the new (reduced) singular points we are going to find the situation represented by Formula (\[tXXxy\]) with non-positive corresponding integers $m,n$. Therefore the preceding strategy of establish the convergence of the natural improper integrals will still be applicable.

Applications to complete vector fields
======================================

Ends of solutions of complete polynomial vector fields on ${{\mathbb C}}^n$
---------------------------------------------------------------------------

This first application concerns Theorem \[introducelabel4\]. Consider a complete polynomial vector field $X$ defined on ${{\mathbb C}}^n$. Set $X=\sum_{i=0}^d X_i$ where $X_i$ stands for the homogeneous component of degree $i$ of $X$. To keep as much as possible the notations used in the previous sections, the foliation associated to $X$ will be denoted by ${\mathcal{D}}$ whereas ${{\mathcal F}}$ will stand for the foliation associated to the top-degree homogeneous component $X_d$. We shall always assume that $d \geq 2$.

Recall that both foliations ${\mathcal{D}}$ and ${{\mathcal F}}$ admit holomorphic extensions to ${{\mathbb C}}P(n)$ and these extensions are also denoted by ${\mathcal{D}}$ and ${{\mathcal F}}$.

\[suzuki1.1\] The homogeneous vector field $X_d$ is not a multiple of the radial vector field $${\mathcal{E}}= x_1 \partial /\partial x_1 + \cdots + x_n \partial /\partial x_n \; .$$

First note that the vector field $X_d$ is semi-complete on all of ${{\mathbb C}}^n$ since it is the top-degree homogeneous component of a complete vector field. Next suppose we have $X_d = P {\mathcal{E}}$ where $P$ is a polynomial of degree $d-1$. The semi-complete character of $X_d$ then implies that $d=2$. Therefore it only remains to exclude this last possibility. For this, note that the generic leaf $L$ of ${\mathcal{D}}$ intersects the hyperplane at infinity of ${{\mathbb C}}P(n)$ transversely at a regular point $p$ for ${\mathcal{D}}$. Besides the point $p$ is regular for the restriction of $X$ to $L$. In other words, the flow of $X$ reaches the hyperplane at infinity in finite time. This is impossible since $X$ is complete on ${{\mathbb C}}^n$. The proof of the lemma is over.

Again let $\Delta_{\infty}$ denote the hyperplane at infinity in ${{\mathbb C}}P(n)$. It follows from the preceding that $\Delta_{\infty}$ is invariant by both ${\mathcal{D}}$ and ${{\mathcal F}}$. Besides the foliations [*induced*]{} on $\Delta_{\infty}$ by ${\mathcal{D}}, \, {{\mathcal F}}$ turn out to coincide. The foliation induced by ${{\mathcal F}}$ on $\Delta_{\infty}$ will be denoted by ${{\mathcal F}}_{\infty}$. Also $\Delta_{\infty}$ corresponds to the divisor of poles for both $X, \, X_d$.

Since ${{\mathcal F}}$ is given by a homogeneous vector field, the ideas developed in Sections 2 and 3 can directly be applied to it. In turn, near $\Delta_{\infty}$, the foliation ${\mathcal{D}}$ becomes very close to ${{\mathcal F}}$. In the sequel we are going to combine these two issues in order to establish Theorem A.

Let us begin by choosing affine coordinates $(x_1, \ldots, x_{n-1}, z)$ analogous to those used in Sections 3, 4. Namely the hyperplane $\{ z=0\}$ is contained in $\Delta_{\infty}$ and the plane at infinity $\Delta_{\infty}^{1,\ldots ,n-1}$ defined by the affine coordinates $x_1, \ldots, x_{n-1}$, where $z=0$ is fixed, is not invariant by the restrictions of either ${\mathcal{D}}$ or ${{\mathcal F}}$ to $\Delta_{\infty}$. We are then able to apply the results of Section 4 to the foliation ${{\mathcal F}}$. In particular, the leaves of ${{\mathcal F}}$ are equipped with the (singular) real foliations ${\mathcal{H}}^{\theta}$ where $\theta$ is chosen in the interval $(-\pi/2 , \pi/2)$. For the rest of this section, these foliations will be denoted by ${\mathcal{H}}_{{{\mathcal F}}}$ (resp. ${\mathcal{H}}^{\theta}_{{{\mathcal F}}}$). To define a suitable version of these real trajectories in the leaves of ${\mathcal{D}}$ we proceed as follows. Given a point $p=
(x_1^0, \ldots, x_{n-1}^0, z^0)$ with $z^0 \neq 0$, let $L_p$ denote the leaf of ${\mathcal{D}}$ through $p$. To define the foliation ${\mathcal{H}}_{{\mathcal{D}}}$ at $p$, we consider the function $(x_1, \ldots, x_{n-1}, z) \mapsto \vert z \vert$ restricted to $L_p$. The tangent vector to ${\mathcal{H}}_{{\mathcal{D}}}$ at $p$ is simply the negative of the gradient of the function in question. Once ${\mathcal{H}}_{{\mathcal{D}}}$ is defined the foliations ${\mathcal{H}}^{\theta}_{{\mathcal{D}}}$ have an obvious definition.

The next step in our construction consists of investigating the basic properties of ${\mathcal{H}}_{{\mathcal{D}}}$ in analogy with the properties of ${\mathcal{H}}_{{{\mathcal F}}}$ considered in Sections 3 and 4. Recalling that ${\mathcal{D}}, \, {{\mathcal F}}$ induce the same foliation ${{\mathcal F}}_{\infty}$ on $\Delta_{\infty}$, consider a point $(x_1^0, \ldots ,x_{n-1}^0, 0) \in \Delta_{\infty}$ that is regular for the restrictions of both ${\mathcal{D}}, \, {{\mathcal F}}$ to $\Delta_{\infty}$. Then we have:

\[suzuki2.2\] The direction of ${\mathcal{H}}_{{\mathcal{D}}}$ at the point $(x_1^0, \ldots ,x_{n-1}^0, z)$ converges uniformly to the direction of ${\mathcal{H}}_{{{\mathcal F}}}$ at $(x_1^0, \ldots ,x_{n-1}^0, 0)$. In particular the foliation ${\mathcal{H}}_{{\mathcal{D}}}$ can be extended to the regular part of ${\mathcal{D}}$ in $\Delta_{\infty}$ and this extended foliation coincides with ${\mathcal{H}}_{{{\mathcal F}}}$ on $\Delta_{\infty}$.

Since the behavior of ${\mathcal{D}}$ near $(x_1^0, \ldots ,x_{n-1}^0, 0)$ is dominated by the component $X_d$ of $X$, it suffices to check that the trajectories of ${\mathcal{H}}_{{{\mathcal F}}}$ admit a definition analogous to the one given above for the trajectories of ${\mathcal{H}}_{{\mathcal{D}}}$. In other words, it suffices to prove that the direction of ${\mathcal{H}}_{{{\mathcal F}}}$ at $(x_1^0, \ldots ,x_{n-1}^0, z)$ coincides with the gradient of the function $(x_1, \ldots, x_{n-1}, z) \mapsto \vert z \vert$ restricted to the leaf of ${{\mathcal F}}$ through $(x_1^0, \ldots ,x_{n-1}^0, z)$. This is however an immediate consequence of Formula (\[omega1\]). The lemma is proved.

To help us to explain how to derive properties of $X, \, {\mathcal{D}}$ from properties of $X_d, \, {{\mathcal F}}$, it is convenient to consider a small neighborhood (in the $n$-dimensional ambient space) $V$ of $({\rm Sing}\, ({\mathcal{D}}) \cap \Delta_{\infty}) \cup {\rm Sing}\, (X)$, where ${\rm Sing}\, ({\mathcal{D}})$ (resp. ${\rm Sing}\, (X)$) stands for the singular set of ${\mathcal{D}}$ (resp. $X$). Next denote by $U$ a neighborhood of $\Delta_{\infty} \setminus V$. Also, in order to keep a “uniform contractive character” over trajectories of ${\mathcal{H}}_{{{\mathcal F}}}^{\theta}$, we fix some (small) $\epsilon >0$ and consider only those values of $\theta$ belonging to the interval $(-\pi/2 + \epsilon, \pi/2 - \epsilon)$.

Note that all the endpoints belonging to $U$ for trajectories of ${\mathcal{H}}_{{{\mathcal F}}}^{\theta}$ are situated over $\Delta_{\infty}^{1,\ldots ,n-1}$. In particular a trajectory of ${\mathcal{H}}_{{{\mathcal F}}}^{\theta}$ through an affine point $(x_1^0, \ldots ,x_{n-1}^0, 0) \in U \cap \Delta_{\infty}$ will never approach $\Delta_{\infty}^{1,\ldots ,n-1}$ unless it first enters $V$. Modulo choosing the neighborhood $U$ sufficiently narrow, the restriction to $U$ of the foliation ${\mathcal{D}}$ is very close to the (restriction to $U$ of the) foliation ${{\mathcal F}}$. A similar statement holds for the foliations ${\mathcal{H}}_{{\mathcal{D}}}$ and ${\mathcal{H}}_{{{\mathcal F}}}$ thanks to Lemma \[suzuki2.2\]. In particular we obtain the following:

\[suzuki3.3\] Let $\epsilon >0$ be fixed. Consider a point $p=(x_1^0, \ldots ,x_{n-1}^0, z^0) \in U$ and denote by $l_p^{+,\theta}$ the semi-trajectory of ${\mathcal{H}}_{{\mathcal{D}}}^{\theta}$ initiated at $p$ for $\theta \in [-\frac{\pi}{2} + \epsilon, \frac{\pi}{2} - \epsilon]$. Consider a path $c :[0,1] \rightarrow  l_p^{+,\theta} \cap U$, with $c(0) =p$, and set $c(1) = (x_1^1, \ldots ,x_{n-1}^1, z^1)$. Then there is a constant ${\rm Cte}$ depending solely on $\epsilon$ such that the following condition is always verified: whenever the length of $c$ exceeds ${\rm Cte}$, we have the estimate $\vert z^1 \vert < \vert z^0 \vert /2$.

It follows immediately from the proof of Theorem \[introducelabel4\] concerning the foliation ${{\mathcal F}}$. More precisely, it was shown that for the analogue statement where the foliation ${\mathcal{H}}_{{\mathcal{D}}}^{\theta}$ is replaced by the foliation ${\mathcal{H}}_{{{\mathcal F}}}$ it suffices to choose ${\rm Cte}$ as $3\ln(2)/2{\alpha}$. The present statement follows from the fact that inside $U$ the foliation ${\mathcal{H}}_{{{\mathcal F}}}$ is “very close” to ${\mathcal{H}}_{{\mathcal{D}}}$.

One last ingredient is still needed for the proof of Theorem A. Let $l_p^{+,\theta}$ be a trajectory as in Lemma \[suzuki3.3\] and denote by $L_p$ the leaf of ${\mathcal{D}}$ containing $l_p^{+,\theta}$. The idea behind the statement of Theorem A consists of estimating the integral of $dT_L$ over $l_p^{+,\theta}$ where $dT_L$ stands for the time-form induced by $X$ on $L_p$. In Section 4 suitable estimates for this type of integral were obtained in the case of homogeneous vector fields. The estimate is based on the “renormalized time-form” induced on $\Delta_{\infty}$ by the vector field and on the evolution of the distance of the points to $\Delta_{\infty}$ (the “height” of the points). As to the height of points, the preceding lemma provides a suitable control of their evolution over trajectories of ${\mathcal{H}}_{{\mathcal{D}}}^{\theta}$ in the case of non-homogeneous polynomial vector fields. Finally we recall that the foliations induced by $X$ and by $X_d$ on $\Delta_{\infty}$ turn out to coincide and the same holds for the “renormalized time-forms” induced on $\Delta_{\infty}$ by $X$ and by $X_d$.

We are now ready to prove Theorem A.

Consider the foliation ${\mathcal{D}}$ induced by $X$ on ${{\mathbb C}}P(n)$ and let $\Delta_{\infty}$ be as above. Let $V$ denote the given neighborhood of $({\rm Sing}\, ({\mathcal{D}}) \cap \Delta_{\infty}) \cup {\rm Sing}\, (X)$ and fix $\epsilon > 0$. Next choose a neighborhood $U$ of $\Delta_{\infty} \setminus V$ so that the statement of Lemma \[suzuki3.3\] is valid. It is sufficient to prove the theorem for the foliation ${\mathcal{H}}_{{\mathcal{D}}}$ since the adaptations needed to the general case of the foliations ${\mathcal{H}}_{{\mathcal{D}}}^{\theta}$, $\theta \in (-\pi/2 + \epsilon , \pi/2 -\epsilon)$, are clear.

Consider a point $p=(x_1^0, \ldots ,x_{n-1}^0, z^0) \in U \setminus V$. Denote by $l_p^+$ (resp. $L_p$) the semi-trajectory of ${\mathcal{H}}_{{\mathcal{D}}}$ initiated at $p$ (resp. the leaf of ${\mathcal{D}}$ through $p$). Suppose first that $l_p^+$ is entirely contained in $U$. To explain the structure of the proof of our theorem, we shall first prove that the preceding assumption contradicts the fact that the vector field $X$ is complete. For this we are going to show that the integral of the time-form $dT_L$ induced by $X$ on $L_p$ over $l_p^+$ is convergent. Since it clearly accumulates on $\Delta_{\infty}$ (in particular $l_p^+$ leaves every compact set contained in $L_p$) the convergence of the mentioned integral contradicts the completeness of $X$. Let us also point out that our claim reduces to Theorem \[introducelabel4\] in the case of homogeneous vector fields.

To adapt the proof of Theorem \[introducelabel4\] to the present case where $X$ is not homogeneous we proceed as follows. The choice of the coordinates $(x_1, \ldots, x_{n-1}, z) = (\underline{x}, z)$ allows us to write $X_d$ in the form $$X_d = z^{1-d} [F_1 (\underline{x}) \partial /\partial x_1 + \cdots + F_{n-1} (\underline{x}) \partial /\partial x_{n-1} + z H (\underline{x}) \partial /\partial z]$$ whereas the vector field $X$ becomes $$X =  z^{1-d}  [ F_1^{\ast} (\underline{x}, z) \partial /\partial x_1 + \cdots + F_{n-1}^{\ast} (\underline{x}, z) \partial /\partial x_{n-1} +
z H^{\ast} (\underline{x}, z) \partial /\partial z \!]  .$$ Besides the coefficients $F_i, \, F_i^{\ast}$, $i=1, \cdots , n-1$, (resp. $H_i, \, H_i^{\ast}$) are related by the formulas $$F_i^{\ast} (x_1, \ldots , x_{n-1}, z) - F_i (x_1, \ldots , x_{n-1}) = z P_i (x_1, \ldots , x_{n-1}, z)$$ (resp. $H^{\ast} (x_1, \ldots , x_{n-1}, z) - H (x_1, \ldots , x_{n-1}) = z Q (x_1, \ldots , x_{n-1}, z)$) where $Q, \, P_i$ are polynomials in the variables in question. Next note that the time-form $dT_L$ is given by $$dT_L = \frac{z^{d-1} }{F_1^{\ast} (x_1, \ldots , x_{n-1}, z)} \, dx_1 = \cdots = \frac{z^{d-1} }{F_{n-1}^{\ast} (x_1, \ldots , x_{n-1}, z)} \, dx_{n-1} \, .$$ Now since $U$ does not intersect the singular set of ${\mathcal{D}}$, we can suppose without loss of generality that $F_1^{\ast} (x_1, \ldots , x_{n-1}, z)$ is bounded from below by a positive constant $\beta$ in $U$, otherwise we replace $F_1^{\ast}$ by a suitable $F_i^{\ast}$. This last estimate combined to Lemma \[suzuki3.3\] then shows that the integral of $dT_L$ over $l_p^+$ is bounded by simply repeating the calculations performed in the proof of Theorem \[introducelabel4\].

We are then led to conclude that the semi-trajectory $l_p^+$ must intersect the neighborhood $V$ of ${\rm Sing}\, ({\mathcal{D}}) \cap \Delta_{\infty}$ regardless of how small is $V$. In particular it may happen that $l_p^+$ accumulates on singular points of ${\mathcal{D}}$ lying in $\Delta_{\infty}$. In this case the integral of $dT_L$ over $l_p^+$ cannot be bounded without additional conditions. Fortunately to establish Theorem A it is not needed to keep track of the amount of “time” that $l_p^+$ spend [*inside*]{} $V$ but rather of the amount of time that $l_p^+$ spend away from $V$. To be more precise let us prove the following:

[*Claim*]{}. The distance between the trajectory $l_p^+$ and the hyperplane $\Delta_{\infty}$ cannot have a minimum unless this minimum is [*zero*]{}. In this latter case the intersection point $l_p^+ \cap \Delta_{\infty}$ is never reached by the flow of $X$.

Before starting the proof of the claim, it is convenient to make some general comments regarding the possibility of having a point $q$ in $l_p^+ \cap \Delta_{\infty}$. A first case where this may happen arises from the definition of “leaf” given in Section 2, borrowed from [@marco2]. According to this definition the leaf $L_p$ of ${\mathcal{D}}$ may contain a singular point of ${\mathcal{D}}$ lying in $\Delta_{\infty}$. In fact, in this case, a local branch of $L_p$ about $q$ defines an irreducible separatrix for ${\mathcal{D}}$ at $q$. It is then natural to think of $q$ as belonging to $l_p^+$. More generally, it may happen that $l_p^+$ converges to a point $q$ lying in $\Delta_{\infty}$ whether or not $q$ belongs to $L_p$. With a small abuse of notation, the point $q$ may be thought of as belonging to $l_p^+$. In all these cases the statement of Theorem A is obvious: the completeness of $X$ implies that the integral of $dT_L$ over a local branch of $l_p^+$ converging to $q$ is infinite. So $l_p^+$ enters every given neighborhood of $q$ and remains inside “forever”. The statement then follows from observing that $q$ must be a singular point of ${\mathcal{D}}$ since $\Delta_{\infty}$ is invariant by ${\mathcal{D}}$.

A further reduction in the proof of Theorem A is possible even though not strictly needed. Namely, with the above notations, we can suppose that (a local branch of) $l_p^+$ never converges to a point $q$ that is singular for $X$ (and in particular for ${\mathcal{D}}$). In fact, if this point belongs $\Delta_{\infty}$ then the theorem results immediately as previously seen. Similarly, if $q \in {\rm Sing}\, (X) \setminus \Delta_{\infty}$, then the theorem follows from the standard results on existence and uniqueness of solutions for regular ordinary differential equations.

[*Proof of the Claim*]{}. Given what precedes, let us suppose for a contradiction that $q$ is a point of minimum for the mentioned distance and that $q$ lies in ${{\mathbb C}}P(n) \setminus \Delta_{\infty}$. First we are going to prove that the point $q$ must belong to the domain of definition of the coordinates $(x_1, \ldots , x_{n-1}, z)$. Since $X$ is not homogeneous and $q$ is not in $\Delta_{\infty}$, this assertion is not an immediate consequence of Lemma \[lineatinfinity\]. Thus, in order to prove it, suppose that $c: [0,1) \rightarrow l_p^+$ is a local parametrization of $l_p^+$ with $\lim_{t\rightarrow 1^-} c(t) =q$. Setting $c(t) = (x_1 (t), \ldots ,x_{n-1} (t), z(t))$, it follows that $z(t)$ is locally bounded at $q$. If, in addition, $(x_1 (t), \ldots ,x_{n-1} (t))$ leaves the domain of definition of coordinates $(x_1, \ldots , x_{n-1}, z)$, then by using standard coordinates of ${{\mathbb C}}P(n)$ whose domain contains $\Delta_{\infty}$, it immediately follows from the bounded character of $z(t)$ that $\lim_{t\rightarrow 1^-} c(t) =q$ actually belongs to $\Delta_{\infty}$. In this case the theorem follows as already shown.

Summarizing the above discussion, we can suppose that $q = (q_1, \ldots, q_n)$ is a regular point for $X, \, {\mathcal{D}}$ belonging to the domain of definition of the coordinates $(x_1, \ldots , x_{n-1}, z)$ and verifying $q_n \neq 0$. A final contradiction can now be obtained as follows. Let $\Phi (T) = (\Phi_1 (T), \ldots , \Phi_n (T))$ be a local parametrization of $L_p$ about $q$ ($\Phi (0) =q$). Since $q$ is a regular point for $X$, the holomorphic map $T \mapsto \Phi_n (T) \in {{\mathbb C}}$ is not constant and hence it has isolated critical points. From the local inversion theorem and the fact that an open set remains open when an [*isolated point of its boundary*]{} is added to it, it follows that $T \mapsto \Phi_n (T)$ is an open map what in turn contradicts the assumption that $\vert \Phi_n \vert$ has a (positive) local minimum at $T=0$. The claim is proved.

To finish the proof of Theorem A consider now the semi-trajectory $l_p^+$. The above discussion shows that $l_p^+$ accumulates on $\Delta_{\infty}$, in particular $l_p^+$ leaves compact sets of $L_p$. The completeness of $X$ then implies $$\int_{l_p^+} dT_L = \infty \; .$$ Consider a decomposition $l_p^+ =  c_1 \ast c_2 \ast \cdots$ of $l_p^+$ in finitely or infinitely many paths such that $c_k$ is contained in $U$ for $k$ odd and $c_k$ is contained in $V$ for $k$ even. The statement is now reduced to prove that $$\sum_{k=0}^{\infty} \left[ \int_{c_{2k+1}} dT_L \right] < \infty \, .$$ The last estimate however follows from the same argument employed above in the case where $l_p^+$ was entirely contained in $U$. It suffices to observe that the claim guarantees that $\vert c_{2(k+1) +1} (0) \vert < \vert c_{2k+1} (1) \vert$. The theorem is proved.

Let us now establish Theorem A’.

Consider again a fixed point $p$ and let $\Phi_p : {{\mathbb C}}\rightarrow L_p$ be given by $\Phi_p (T) = \Phi (T, p)$ where $L_p$ stands for the leaf of ${\mathcal{D}}$ through $p$. In the affine coordinates $(x_1, \ldots ,x_{n-1}, z)$ the map $\Phi_p$ becomes $(\Phi_1 (T), \ldots , \Phi_n (T))$. In particular this allows us to define the Abelian form $\eta$ on ${{\mathbb C}}$ by letting $\eta = - \Phi_n^{'} dT / \Phi_n$. Next, if the oriented foliation ${\mathcal{H}}$ is restricted to $L_p$, then we can consider the corresponding pulled-back oriented foliation $\Phi_p^{\ast} {\mathcal{H}}$ on ${{\mathbb C}}$. In this direction we have:

[*Claim 1*]{}: The oriented foliation $\Phi_p^{\ast} {\mathcal{H}}$ coincides with the real (positive) foliation induced by $\eta$.

[*Proof of Claim 1*]{}. Consider a point $q = \Phi_p (T_0) \in L_p$ that is regular for ${\mathcal{H}}$. The direction of ${\mathcal{H}}$ at $q$ is determined by the negative of the gradient of the “height” function $(x_1, \ldots ,x_{n-1}, z) \mapsto \Vert z \Vert$ restricted to $L_p$. In the coordinate $T$ this function is simply $T \mapsto \Vert \Phi_n (T) \Vert$. The gradient direction of this latter function is determined by the condition that $\Phi_n^{'} (T_0) (T-T_0)$ must be aligned with $\Phi_n (T_0)$. This amounts to saying that the direction of $\Phi_p^{\ast} {\mathcal{H}}$ at $T_0$ is nothing but the positive real direction of $\eta$.

To abridge notations the foliation $\Phi_p^{\ast} {\mathcal{H}}$ will be denoted by $\{ {\rm Arg}\, \eta =0\}$. More generally the pull-back by $\Phi_p$ of the foliations ${\mathcal{H}}^{\theta}$ coincide with $\{ {\rm Arg}\, \eta =\theta\}$, in particular $\{ {\rm Arg}\, \eta =\pi/2\}$ is the foliation orthogonal to $\{ {\rm Arg}\, \eta =0\} = \Phi_p^{\ast} {\mathcal{H}}$.

The separating curve $c_0$ to be chosen is given by the trajectory of $\{ {\rm Arg}\, \eta =\pi/2\}$ through $T_0$ i.e. a “leaf” of ${\mathcal{H}}^{\theta}$ for $\theta = \pi/2$. Geometrically $\Phi_p (c_0)$ is the curve determined in $L_p$ by the intersection of $L_p$ itself with the real hypersurface $\{ \Vert z \Vert =
\vert \Phi_n (T_0) \vert \}$. This curve may be closed. Next we choose the component $\mathcal{U}^+$ of ${{\mathbb C}}\setminus c_0$ that corresponds to the saturated of $T_0$ by the spray of trajectories of $\{ {\rm Arg}\, \eta =\theta\}$ issued from $T_0$ with $\theta \in (-\pi/2 ,\pi/2)$. To check that $\mathcal{U}^+$ is unbounded just notice that a trajectory $l_p^{+, \theta} \subset L_p$, $\theta \in (-\pi/2 ,\pi/2)$, issued from $p$ will, by construction, accumulate on $\Delta_{\infty}$ unless it accumulates on a singularity of $X$ (in ${{\mathbb C}}^n$). The statement being clear in the latter case, let us consider that $l_p$ accumulates on $\Delta_{\infty}$. In particular it leaves every compact set contained in $L_p$. Since $X$ is complete, it results that the integral of the corresponding time-form over $l_p^{+, \theta}$ is unbounded. Next note that the preimage of $l_p^{+, \theta}$ by $\Phi_p$ is the trajectory of $\{ {\rm Arg}\, \eta =\theta\}$ issued from $T_0$. Furthermore the preimage by $\Phi_p$ of the time-form induced on $L_p$ is nothing but the canonical form $dT$ on ${{\mathbb C}}$. Thus the integral of the time-form over segments of $l_p^{+, \theta}$ is equal to the integral of the form $dT$ over corresponding segments of the mentioned trajectory. It then follows that the trajectory in question must leave every compact set contained in ${{\mathbb C}}$ what shows that $\mathcal{U}^+$ is unbounded.

Summarizing to show that $\mathcal{U}^+$ satisfies all the conditions in the statement it only remains to check that $$\lim_{r \rightarrow \infty} \frac{ {\rm Meas}\, (\mathcal{T}_V \cap B_r)}
{{\rm Meas}\, (\mathcal{U}^+ \cap B_r)} =1 \, . \label{thelastone}$$

To begin with, note that $\eta$ is holomorphic in $\mathcal{U}^+$ since $\Phi_n (T)$ never reaches $0 \in {{\mathbb C}}$. Furthermore the trajectories of ${\mathcal{H}}^{\theta}$, $\theta \in (-\pi/2 ,\pi/2)$, approach $\{ z=0\}$. These trajectories, in fact, remain in a compact part of the domain of definition of the coordinates $(x_1, \ldots ,x_{n-1}, z)$ since the “infinity” of $\{ z=0\}$ consists of poles with residue equal to $1$ for the abelian form $\omega_1$ in Section 3, cf. Lemma \[lineatinfinity\]. The existence of poles with residue equal to $1$ implies that on a neighborhood of this “plane at infinity” the trajectories of $\{ {\rm Arg}\, \eta =\pi/2\}$ are closed curves and the trajectories of ${\mathcal{H}}^{\theta}$ for $\theta \in (-\pi/2 ,\pi/2)$ point outward these closed curves. The preceding claim then becomes clear. As a consequence of this, we conclude that the absolute value of the coefficient of $\eta$ is uniformly bounded in $\mathcal{U}^+ \setminus \mathcal{T}_V$ since away from $\mathcal{T}_V$ we can choose the denominator of $\omega_1 = -H/F$ to be bounded from below by a positive constant. Next observe that $\eta$ defines a singular flat structure on $\mathcal{U}^+$ and for this flat structure the trajectories of $\{ {\rm Arg}\, \eta =\theta\}$ are geodesics (“straight lines”). This leads us to

[*Claim 2*]{}: Given $\varepsilon >0$, there is $\delta >0$ such that the saturated $\mathcal{U}_{\delta}^+$ of $T_0$ by trajectories of $\{ {\rm Arg}\, \eta =\theta\}$ for $\theta \in (-\pi/2 +\delta ,\pi/2 - \delta)$ verifies $$\liminf_{r \rightarrow \infty}  \frac{ {\rm Meas}\, [(\mathcal{U}_{\delta}^+ \cap B_r) \cup (\mathcal{T}_V \cap (\mathcal{U}^+ \setminus \mathcal{U}^+_{\delta}))]}
{{\rm Meas}\, (\mathcal{U}^+ \cap B_r)} > 1- \varepsilon \, .$$

[*Proof of Claim 2*]{}. The statement would be clear if the flat structure induced by $\eta$ were the standard flat structure of ${{\mathbb C}}$. Consider an arc of circle $S_{r_0}$ (about $T_0$) of radius $r_0$ whose interior contains no singular points of $\eta$. Consider also two trajectories $l_{\theta_1}, \, l_{\theta_2}$ issued from $T_0$. Since $X$ is complete these trajectories intersect $S_{r_0}$ at points $T_1, T_2$. Because $\eta$ is closed, the integral of $\eta$ over the boundary of the triangle whose sides are the segments of $l_{\theta_1}, \, l_{\theta_2}$ delimited by $T_0$ and $T_1, T_2$ and the corresponding arc of $S_{r_0}$ determined by $T_1, T_2$ equals zero. Finally since the coefficient of $\eta$ is uniformly bounded (and bounded from below if we stay away from its singular points), we conclude that the length of the arc of $S_{r_0}$ determined by $T_1, T_2$ is bounded by ${\rm Const} r_0 \vert \theta_1 - \theta_2 \vert$ for every pair $\theta_1, \theta_2$. From this we obtain the desired estimate provided that $S_{r_0}$ contains no singularities of $\eta$ in its interior.

To finish the proof of the claim, we need to consider the effect of the singularities of $\eta$. These singularities are of saddle type since $\eta$ is holomorphic on $\mathcal{U}^+$. For every singular point of $\eta$ we consider a disc of uniform radius about the corresponding point in $L_p$. In the complement of the union of these discs $\omega_1$, and therefore $\eta$ in the coordinate $T$, is bounded from below by a positive constant. The claim will be proved if the union of these discs in the coordinate $T$ has area less than $\varepsilon r/2$ for $r$ large. In fact, in the complement of this union $\eta$ is bounded from below by a positive constant and from above by the previous constant so that the preceding argument can be applied in finitely many regions of a ball $B_r$. Finally to justify the previous claim note that, in order to prove the desired estimate, we only need to consider those discs about points in $L_p$ that lie in the complement of $V$. It then follows that in these discs the norm of $X$ is bounded from below by a positive constant so that their size in the coordinate $T$ is uniformly bounded. Besides the distance in the leaf $L_p$ between every two discs as before is bounded from below by a positive constant. Though this property is not directly reflected in the coordinate $T$ since the norm of $X$ increases (i.e. $X$ becomes “faster”), the size of the corresponding neighborhoods reduces proportionally to the increase of the norm of $X$. This quickly leads to the desired conclusion and establishes the claim.

In view of Claim 2 to end the proof of Theorem A’ it suffices to show that $$\lim_{r \rightarrow \infty} \frac{ {\rm Meas}\, (\mathcal{T}_V \cap B_r \cap \mathcal{U}_{\delta}^+)}
{{\rm Meas}\, (\mathcal{U}_{\delta}^+ \cap B_r)} =1 \, ,$$ for fixed positive $\delta$. To do this consider $r$ given. Next observe that every for $\theta \in (-\pi/2 +\delta ,\pi/2 - \delta)$ the corresponding trajectory $l_{\theta}$ of $\{ {\rm Arg}\, \eta =\theta\}$ issued from $T_0$ intersects the boundary $\partial B_r$ of $B_r$ since $X$ is complete. Let $T_{\theta, r}$ be this intersection point and denote by $l_{\theta, r}$ the segment of $l_{\theta}$ delimited by $T_0$ and $T_{\theta, r}$. According to Theorem A, there is uniform constant ${\rm Cte}$ (depending neither on $\theta$ nor on $r$) such that the length of the segments of $l_{\theta, r}$ corresponding to those instants where $\Phi (T)$ remains away from $V$ is bounded by ${\rm Cte}$ whereas the length of $l_{\theta, r}$ goes to infinity as $r \rightarrow \infty$. The statement of Theorem A’ now results from a standard application of Fubini’s theorem.

Complete polynomial vector fields on ${{\mathbb C}}^n$ with simple singularities at infinity
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In this section we shall give an application of Theorem \[maintheo\] that cannot be obtained from Theorem \[introducelabel4\] alone. Let $X$ be a complete polynomial vector fields on ${{\mathbb C}}^n$ and denote by ${\mathcal{D}}$ its associated foliation. Recall that we make no distinction between ${\mathcal{D}}$ viewed as a foliation on ${{\mathbb C}}^n$ and ${\mathcal{D}}$ viewed as a foliation on ${{\mathbb C}}P(n)$. Again $X_d$, $d \geq 2$, will denote the homogeneous component of highest degree of $X$. The foliation associated to $X_d$ is denoted by ${{\mathcal F}}$ and can also be viewed as a foliation on both ${{\mathbb C}}^n$ or ${{\mathbb C}}P(n)$. Recall that the singularities of ${\mathcal{D}}$ in the hyperplane at infinity $\Delta_{\infty}$ are “simple” in the sense of Conditions 1 and 2 given in the Introduction (just before the statement of Theorem B). It then follows that they are isolated inside $\Delta_{\infty}$ (but maybe not inside ${{\mathbb C}}P(n)$). Furthermore note that if a non-constant polynomial $P$ divides $X_d$ it must also divide $X$. Otherwise the curve induced on $\Delta_{\infty}$ by $\{ P=0\}$ would contain singularities of ${\mathcal{D}}$ whose linear part is degenerate what is impossible. In turn this implies that the singular sets of ${\mathcal{D}}$ and of ${{\mathcal F}}$ coincide on $\Delta_{\infty}$. Furthermore $X_d$ must have a singular set of codimension at least $2$ since the singular set of $X$ has codimension two or greater.

Let us also observe that the restriction of ${\mathcal{D}}$ to $\Delta_{\infty}$ coincides with the foliation ${{\mathcal F}}_{\infty}$ induced by ${{\mathcal F}}$. Also if $q\in \Delta_{\infty}$ is a (necessarily common) singular point of ${\mathcal{D}}, \, {{\mathcal F}}$, then the corresponding linear parts of these foliations at $q$ turn out to coincide.

\[Cstar1.1\] With the definition of leaf given in Section 3 (borrowed from [@marco2]), the leaves of ${{\mathcal F}}_{\infty}$ are either rational curves or Riemann surfaces uniformized by ${{\mathbb C}}$.

We need to show that the leaves of ${{\mathcal F}}_{\infty}$ cannot be hyperbolic Riemann surfaces. Since ${{\mathbb C}}P(n)$ has a Kähler structure, it follows from the main result of [@marco2] that the set of parabolic leaves of ${\mathcal{D}}$ is a pluri-polar set unless it coincides with the whole space. Since the leaves of ${\mathcal{D}}$ contained in ${{\mathbb C}}^n$ are clearly parabolic, we conclude that the leaves of ${\mathcal{D}}$ contained in $\Delta_{\infty}$ must also be parabolic (or rational). The latter leaves however are precisely the leaves of ${{\mathcal F}}_{\infty}$.

The next step is to consider the foliation ${{\widetilde{\mathcal F}}}$ induced on the manifold $M$ by ${{\mathcal F}}$ as in Sections 3, 4 and 5. In particular the foliation ${{\widetilde{\mathcal F}}}_{\infty}$ is naturally identified with ${{\mathcal F}}_{\infty}$. In view of the existence of the projections ${{\mathcal P}}_0, \, {{\mathcal P}}_{\infty}$ introduced in Section 3, Lemma \[Cstar1.1\] implies that no leaf of ${{\widetilde{\mathcal F}}}$ is hyperbolic. Note that this conclusion cannot directly be derived from the vector field $X_d$ since $X_d$ need not be complete (though it is semi-complete).

The next proposition is crucial for the proof of Theorem B. In turn its proof relies heavily on Theorem \[maintheo\].

\[hyperbolicleaves\] There exists a singularity of ${{\widetilde{\mathcal F}}}_{\infty}$ providing a sink singularity for ${\mathcal{H}}$ (resp. ${\mathcal{H}}^{\theta}$) restricted to a generic leaf of ${{\widetilde{\mathcal F}}}_{\infty}$.

Let us suppose for a contradiction that the statement is false. In view of the material in Section 3, it follows that the trajectories of ${\mathcal{H}}$ have no future endpoint. Therefore these trajectories have all infinite length and this will be exploited in the proof. To begin with note that $L_q$ is not contained in a rational curve otherwise ${{\widetilde{\mathcal F}}}_{\infty}$ would be a rational pencil on $\Delta_{\infty}$. In this case the foliation ${\mathcal{H}}$ would have sink singularities when restricted to a generic leaf of ${{\widetilde{\mathcal F}}}_{\infty}$.

Our strategy to obtain a contradiction will consist of showing that a “generic” leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ cannot be a “parabolic” (i.e. non-hyperbolic) Riemann surface. For this consider $q \in \mathcal{P}^{-1}_{\infty}(p)$ as in the statement of Theorem \[maintheo\] and denote by $L_q$ the leaf of ${{\widetilde{\mathcal F}}}$ through $q$. In particular ${{\mathcal P}}_{\infty} (L_q) =L_{\infty}$. Now note that ${{\widetilde{X}}}_d$ is regular over $L_q$. Since ${{\widetilde{X}}}_d$ is semi-complete, this leaf is recovered by an open set ${\bf U} \subseteq {{\mathbb C}}$ which is the domain of definition of the semi-complete flow on $L_q$. In fact, the assumption of having a semi-complete vector field is equivalent to saying that ${\bf U}$ is a maximal domain of definition for the solution $\phi$ of the equation associated to this vector field with initial condition at $q$. Now the lift of $l_q$ in ${\bf U}$ has finite length for the natural euclidean metric of ${{\mathbb C}}$. Indeed, this length is nothing but the integral $\int_{l_q} dT$. Hence the mentioned lift converges to a point $t_0$ in the boundary of ${\bf U}$.

Recalling that $L_q$ must be a non-hyperbolic Riemann surface, and noting that it cannot be contained in a rational curve, it follows that ${\bf U} = {{\mathbb C}}\setminus \{ t_0\}$. In particular, $L_q$ cannot be compactified in an elliptic curve. Thus $L_q$ is actually not contained in any compact curve. Consider now the plane ${{\mathbb C}}$ equipped with the coordinate $t$. In particular the canonical form $dt$ coincides with the pull-back by $\phi$ of the time-form induced on $L_q$ by $X$. Also denote by ${\mathcal{H}}_t$ the lift of the foliation ${\mathcal{H}}$ (restricted to $L_q$) to the ${{\mathbb C}}$-plane. ${\mathcal{H}}$ is naturally a foliation defined on ${\bf U} = {{\mathbb C}}\setminus \{ t_0\}$. Yet we have the following claim whose meaning is fully explained below.

[*Claim*]{}. The point $t_0$ represents a “sink singularity” for ${\mathcal{H}}_t$.

[*Proof of the Claim*]{}. Clearly $L_q$ possesses a cylindrical end. In other words, the map $\phi$ allows us to realize a punctured neighborhood of $t_0 \in {{\mathbb C}}$ as an end of $L_q$. Next consider a point $q' = \phi (t')$ for $t'$ near $t_0$. The trajectory of ${\mathcal{H}}$ through $q'$ is infinite and thus the integral of the corresponding time-form over this trajectory must converge again to a point lying in the boundary of ${\bf U} = {{\mathbb C}}\setminus \{ t_0\}$. We then conclude that the mentioned integral converges to $t_0$. In the ${{\mathbb C}}$-plane equipped with the coordinate $t$, the preceding translates into the fact that the integral of the canonical form $dt$ over the lift (by $\phi$) of the ${\mathcal{H}}$-trajectory through $q'$ converges to $t_0$. This lift however is nothing but the trajectory of ${\mathcal{H}}_t$ through $t'$, it then follows that the trajectory in question converges to $t_0$ and this is the contents of the claim.

The above proof needs further comments. As mentioned ${\mathcal{H}}_t$ is the foliation induced on the plane ${{\mathbb C}}$ by the $1$-form $\phi^{\ast} \omega_1$ which is meromorphic on ${\bf U} = {{\mathbb C}}\setminus \{ t_0\}$ but not necessarily at $t_0$. Indeed, $t_0$ may be an essential singularity for $\phi^{\ast} \omega_1$. Previously in our discussion of singularities for this type of foliation, the “classification” as saddle, source or sink applied only for meromorphic forms (often quoted as abelian forms) and therefore they need not apply to essential singular points of the corresponding forms. This is why the expression sink singularity was written between quotes in the statement of the claim. Whereas no general discussion of singularities obtained at essential singular points of $1$-forms was carried out, the preceding proof shows that $t_0$ topologically looks like a sink for ${\mathcal{H}}_t$ in the sense that ${\mathcal{H}}_t$-trajectories through nearby points converge to $t_0$. To avoid confusion, this type of situation will be referred to from now on as constituting an [*improper sink*]{} for ${\mathcal{H}}$. Thus in the “time coordinate” $t$, the points $t_0$ is viewed as a “topological sink” for ${\mathcal{H}}_t$ (or for ${\mathcal{H}}$) in the sense discussed above. Nonetheless if we look at the actual ${\mathcal{H}}$-trajectories in the leaf $L_q$, those trajectories previously viewed as “converging to $t_0$” become trajectories of ${\mathcal{H}}$ having infinite length.

Summarizing what precedes, the foliation ${\mathcal{H}}_t$ on the plane ${{\mathbb C}}$ has a unique improper sink singularity, corresponding to $t=t_0$, and no (ordinary) sink singularity. Furthermore all trajectories of ${\mathcal{H}}_t$ converge to $t_0$. Indeed, the integral of the time-form over every trajectory of ${\mathcal{H}}$ converges to a point in the boundary of ${\bf U} = {{\mathbb C}}\setminus \{ t_0\}$ and hence to $t_0$ itself.

To finish the proof of the lemma, we are going to show that the situation described above cannot happen. In fact, let $t^{\ast}_1$ be a [*source singularity of ${\mathcal{H}}_t$*]{}. Note that $t^{\ast}_1$ exists since this type singularity is produced by the intersections of ${{\mathcal P}}_{\infty} (L_q)$ with the hyperplane at infinity $\Delta_{\infty}^{(x,y)}$ in the coordinates $(x,y)$ (i.e. the hyperplane at infinity corresponding to affine coordinates for $\Delta_{\infty}$). Since $\Delta_{\infty}^{(x,y)}$ neither is invariant by ${{\widetilde{\mathcal F}}}_{\infty}$ nor there are singularities of ${{\widetilde{\mathcal F}}}_{\infty}$ lying in $\Delta_{\infty}^{(x,y)}$, it follows that every leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ intersects non-trivially $\Delta_{\infty}^{(x,y)}$. These intersections produce then source singularities for ${\mathcal{H}}_t$. These source singularities also have residue equal to $1$.

Next note that away from saddle-singularities of ${\mathcal{H}}_t$, this foliation can be given a structure of transverse riemannian foliation: just parametrize the trajectories of the orthogonal foliation by the integral of $\phi^{\ast} \omega_1$. Thus there is a region $I_1^{\ast}$ on a small loop $c$ about $t_0$ such that the integral of $\phi^{\ast} \omega_1$ over $I_1^{\ast}$ equals $-1$ (the negative of the residue of $\phi^{\ast} \omega_1$ at $t^{\ast}_1$). In fact, $I_1^{\ast}$ is obtained by the intersections with $c$ of the trajectories emanated from $t^{\ast}_1$ (recalling that they all converge to $t_0$). However, $c$ can be chosen so that $\phi^{\ast} \omega_1$ is bounded on a neighborhood of $c$ and hence the integral of $\phi^{\ast} \omega_1$ over $c$ is well-defined. To derive a final contradiction, it suffices to observe that in the complement of the small disc bounded by $c$, there are infinitely many source singularities $t^{\ast}_1, t^{\ast}_2, \ldots$ for ${\mathcal{H}}_t$. In fact, under this assumption, each of these singularities determine as above a region $I_k^{\ast}$ over which the integral of $\phi^{\ast} \omega_1$ equals $-1$. Finally all these regions $I_1^{\ast}, I_2^{\ast}, \cdots$ are clearly pairwise disjoint what immediately leads to a contradiction with the existence of a bound for $\phi^{\ast} \omega_1$ on a neighborhood of $c$.

Finally it only remains to check the existence of infinitely many source singularities $t^{\ast}_1, t^{\ast}_2, \ldots$ in the complement of the disc (containing $t_0$) and bounded by $c$. Let us then suppose that the number of the mentioned source singularities is finite. It then follows that the solution $\phi$ is bounded in the complement of a compact region of the plane ${{\mathbb C}}$. Hence this solution can be extended to a neighborhood of the infinity in ${{\mathbb C}}$. Now the image of $\infty$ in ${{\mathbb C}}\cup \{ \infty \} \setminus \{ t_0 \}$ under the extension of $\phi$ can be nothing but a singular point of the complete vector field $X$ on the affine ${{\mathbb C}}^3$. Modulo using the standard Remmert-Stein theorem, it follows that $L_q$ contains a local separatrix for an affine singularity $p$ of $X$. In other words, a generic leaf of ${{\widetilde{\mathcal F}}}$ defines a local separatrix of ${{\widetilde{\mathcal F}}}$ at one of its singular points. This possibility can be ruled out since the statement of Theorem B would follow immediately from considering the restriction of $X$ to these separatrices on a neighborhood of the corresponding singular points (where $X$ is actually holomorphic). We are then left with a contradiction that finishes our proof.

For the purpose of proving Theorem B, Proposition \[hyperbolicleaves\] can be summarized by saying that there is a singularity $q \in \Delta_{\infty}$ of ${{\widetilde{\mathcal F}}}$ all of whose eigenvalues $\lambda^q_1, \ldots, \lambda^q_n$ belong to ${{\mathbb R}}_+^{\ast}$. This assertion can, in turn, slightly be improved. In fact, recall that $Q . Y_1$ is a semi-complete vector field where $Y_1$ is a linear vector field with real positive eigenvalues $\lambda^q_1, \ldots , \lambda^q_n$. Suppose first that $Q$ is a rational function whose both pole divisor and zero divisor contain the origin. A simple analysis of the possibilities shows that under these conditions we must have $\{ \lambda^q_1, \ldots , \lambda^q_3 \} \subset {{\mathbb Z}}_+$. The other possibility for $Q$ is that it has a (unique) polar component of degree $d-1$ passing through $q$ (actually coinciding with $\Delta_{\infty}$) and empty zero divisor. In this case the foliation ${{\widetilde{\mathcal F}}}$ has a smooth separatrix transverse to $\Delta_{\infty}$. By restricting $X_d$ to this separatrix the semi-complete condition implies that $d$ must be equal to $2$. Besides the local holonomy of the separatrix in question must be trivial (cf. for example [@guillotThesis]). It then follows that each of the eigenvalues $\lambda^q_1 ,\ldots , \lambda^q_{n-1}$ is a multiple of the eigenvalue $\lambda^q_n$. Hence we can set $\lambda^q_n =1$ and $\lambda^q_1 , \ldots , \lambda^q_{n-1} \in {{\mathbb Z}}_+$. This refined statement will lead us to

\[Cstar3.3\] The set formed by the separatrices of ${\mathcal{D}}$ at $q$ contains non-trivial open sets of ${{\mathbb C}}^n$.

Consider the vector field $X$ (resp. foliation ${\mathcal{D}}$) on a neighborhood of the singularity $q \in \Delta_{\infty}$. What precedes ensures us that the $n$ eigenvalues $\lambda^q_1, \ldots , \lambda^q_3$ of ${\mathcal{D}}$ at $q$ are strictly positive integers. Therefore they belong to the Poincaré domain. In particular the corresponding local vector field is either linearizable or it admits a Poincaré-Dulac normal form. Since the latter possibility was ruled out by assumption, our local vector field must be linearizable at $q$. The statement follows now from the fact that $\{ \lambda^q_1, \ldots , \lambda^q_3 \} \subset {{\mathbb Z}}_+$.

Consider a local leaf of ${\mathcal{D}}$ defining a separatrix $\mathcal{S}$ for ${\mathcal{D}}$ at $q$ as above. Since $\mathcal{S}$ may be singular at $q$, we consider also a local irreducible Puiseux parametrization $\gamma (t)$ for $\mathcal{S}$ where $t$ is defined on a neighborhood of $0 \in {{\mathbb C}}$ and $\gamma (0) =q$. Since $\mathcal{S}$ is invariant by $X$, we consider the pull-back $\gamma^{\ast} X$ by $\gamma$ of the restriction of $X$ to $\mathcal{S}$. The irreducible character of $\gamma$ ensures that $\gamma^{\ast} X$ is holomorphic and semi-complete on a neighborhood of $0 \in {{\mathbb C}}$. Therefore the order of $\gamma^{\ast} X$ at $0 \in {{\mathbb C}}$ belongs to the finite set $\{ 0, 1, 2\}$. Furthermore, we can actually exclude the case in which this order equals zero since otherwise points of $L$ would reach $q \in \Delta_{\infty}$ in finite time and this contradicts the fact that $X$ is complete on ${{\mathbb C}}^n$.

Suppose now that the order of $\gamma^{\ast} X$ is $2$. Modulo adding the point $q$ to the global leaf $L$, it follows that $L$ is a Riemann surface equipped with a complete holomorphic vector field having a quadratic singularity. Thus $L$ must be the Riemann sphere and hence (of closure) compact. This being valid for set of leaves of ${\mathcal{D}}$ containing open sets of ${{\mathbb C}}^n$ (cf. Lemma \[Cstar3.3\]) it follows that ${\mathcal{D}}$ defines a rational pencil on ${{\mathbb C}}^n$ (and in particular it is completely integrable).

Finally consider the case where the order of $\gamma^{\ast} X$ at $0 \in {{\mathbb C}}$ equals $1$. In this case the time-form induced on $L$ by $X$ has a non-trivial period. Besides the restriction of $X$ to the (normalization of) the closure of $L$ contains a singularity so that the (closure of) $L$ cannot be an elliptic curve. It then follows that $L$ cannot be an orbit of type ${{\mathbb C}}$. Now it follows from Lemma \[Cstar3.3\] and from Suzuki’s theorem [@suzuki] that the generic orbit of $X$ has type ${{\mathbb C}}^{\ast}$. A theorem due to Brunella then guarantees the existence of a non-constant first integral $R$ for ${\mathcal{D}}$, cf. [@marco1].

To conclude that $X$ is completely integrable, i.e. that ${\mathcal{D}}$ is a rational pencil we shall proceed by induction on the dimension $n$. Indeed it will suffice to check the theorem for $n=3$ since the induction step will then become clear. Therefore let us fix a point $p \in {{\mathbb C}}^n$ that is regular for $X$. Denote by $L_p$ the leaf of ${\mathcal{D}}$ through $p$ and let $S$ stand for the surface level of $R$ containing $p$. The preceding argument also holds for the restriction of $X$ to $S$. In fact, consider the foliation ${\mathcal{D}}_S$ obtained on $S$ by restricting ${\mathcal{D}}$. The assumptions made about the structure of the singularities of ${\mathcal{D}}$ lying in $\Delta_{\infty}$ automatically implies similar assumptions for the singularities of ${\mathcal{D}}_S$ lying in (the desingularization of) $S \cap \Delta_{\infty}$. In other words, we can assume without loss of generality that the “generic” orbit of $X$ [*restricted to the affine part of $S$*]{} still is of type ${{\mathbb C}}^{\ast}$. In fact the results of Suzuki are valid for Stein manifolds. Similarly the above mentioned theorem of Brunella also holds for the restriction of $X$ to $S$. It means that ${\mathcal{D}}_S$ also possesses a non-constant rational first integral $R_2$. However, for $n=3$, $S$ is an algebraic surface of dimension $2$. The existence of $R_2$ then implies that the restriction of ${\mathcal{D}}$ to $S$ is a pencil in a natural sense. Thus the closure of $L_p$ is, in fact, an algebraic curve (possibly singular). Because the restriction of $X$ to this compact curve contains a singular point it must be a rational curve, modulo normalization. Summarizing the closure of $L_p$ is a (possibly singular) rational curve. The theorem is proved when $n=3$. The general case follows easily by induction.

\[tobeincluded\]

Throughout this paper, the affine coordinates $(x,y,z)$ (as well as their higher dimensional versions) used in the construction of $M$ where generically chosen in the sense that $\Delta_{\infty}^{(x,y)}$ remained away from the singularities of ${{\widetilde{\mathcal F}}}_{\infty}$. In particular, in the context of Theorem B, this was useful to establish Proposition \[hyperbolicleaves\] guaranteing the existence of a singular point all of whose eigenvalues are strictly positive. This argument will be repeated in the context of Theorem C to be proved in Section 7.1. The point we want to make here is that further information, such as the existence of a second singularity with similar property, may be obtained by choosing the coordinates $(x,y,z)$ so as to make $\Delta_{\infty}^{(x,y)}$ to pass through the first singularity having only strictly positive eigenvalues.

The above mentioned choice of coordinates $(x,y,z)$ is particularly useful because, under very general conditions, the “source character” of points lying in $\Delta_{\infty}^{(x,y)}$ may cancel out the “sink character” of the singularity in question. In other words, we can avoid the singularity fixed in the beginning of producing “future endpoints” for the trajectories of ${\mathcal{H}}$. If we know that the leaves of ${{\widetilde{\mathcal F}}}_{\infty}$ are parabolic Riemann surfaces, it then follows the existence of some other point in $\Delta_{\infty}$ yielding a sink singularity for ${\mathcal{H}}$

Theorem C and Halphen vector fields
===================================

This section contains the proof of Theorem C along with a discussion of some of the results obtained in [@guillotIHES] with proofs obtained through the methods developed in the course of this work. The results presented here about Halphen vector fields are not new and often less precise than the original theorems of A. Guillot so that they can be regarded simply as further illustrations of our techniques. Nonetheless the setting introduced here makes sense for much more general vector fields transverse to singular fibrations. In particular we plan to apply similar ideas to some Painlevé equations as well as to other classical equations including several Chazy’s equations. This discussion will also motivate a conjecture that would greatly enhance the applicability of all these ideas.

Guillot’s lattices of quadratic vector fields : examples and results
--------------------------------------------------------------------

This paragraph concerns an application of our techniques to the work of A. Guillot in [@guillotFourier], [@guillotIHES] (cf. also [@guillotThesis] for an introduction to both papers). Let us place ourselves in the context of [@guillotFourier]. We are therefore dealing with semi-complete (homogeneous) quadratic vector fields $X^2$ with isolated singularities defined on ${{\mathbb C}}^{n}$. The foliation ${{\widetilde{\mathcal F}}}$ associated to $X^2$ on $M$ leaves $\Delta_{\infty}$ (resp. $\Delta_0$) invariant. The pole divisor (resp. zero divisor) of the lift to $M$ of $X^2$ consists of $\Delta_{\infty}$ (resp. $\Delta_0$) with multiplicity exactly $1$. Finally ${{\widetilde{\mathcal F}}}$ has exactly $2^n -1$ singularities on $\Delta_{\infty}$ (resp. $\Delta_0$) and all of them possess $n$ eigenvalues different from [*zero*]{}.

Let us denote by $p_1, \ldots , p_{2^n -1}$ (resp. $q_1, \ldots , q_{2^n -1}$) the singularities of ${{\widetilde{\mathcal F}}}$ in $\Delta_0$ (resp. their dual singularities in $\Delta_{\infty}$). Following [@guillotFourier], [@guillotThesis] convention, the eigenvalues of ${{\widetilde{\mathcal F}}}$ at $p_i$ are $1, u_1^i , \ldots , u_{n-1}^i$ where $1$ is the eigenvalue corresponding to the radial direction. According to Guillot, for a semi-complete vector field $X^2$ as above, $u_1^i, \ldots , u_n^i$ are all integers. Besides setting $\xi_i = u_1^i \times \cdots \times u_{n-1}^i$, $X^2$ yields a solution for the equation $$\sum_{i=1}^{2^n-1} \frac{1}{\xi_i} = (-1)^{n+1} \, . \label{egyptianfraction}$$ In particular the problem of precisely classifying vector fields as above is naturally related to these egyptian fractions. In general this problem is quite intricate as attested by the multitude of interesting examples presented in the above mentioned works. Our contribution to this problem begins with the following lemma:

\[Guillotlattice1.1\] The leaves of a vector field in Guillot lattice having no dicritical singularity at infinity are hyperbolic Riemann surfaces (where by dicritical it is meant that all the $n$ eigenvalues of the singularity are positive real).

If there is no dicritical singularity in $\Delta_{\infty}$, there cannot exist a future endpoint for the trajectories of ${\mathcal{H}}, \, {\mathcal{H}}^{\theta}$, $\theta \in (-\pi/2, \pi/2)$. In other words, the trajectories of the foliations $\mathcal{H}^{\theta}$ must be infinite. Thus we can apply Theorem \[maintheo\] and the argument of Proposition \[hyperbolicleaves\] to conclude the statement.

The converse of Lemma \[Guillotlattice1.1\] does not hold. A particularly interesting example being provided by Halphen vector fields (defined on ${{\mathbb C}}^3$). The dynamics and geometry of these vector fields are beautifully described in [@guillotIHES], the reader is referred to this paper for background information. From our point of view, these vector fields provide an interesting example illustrating some aspects of our discussion.

First we note that Halphen vector fields possess dicritical singularities in $\Delta_{\infty}$ (as well as in $\Delta_0$). Yet, under natural additional conditions, they are semi-complete and possess leaves that are isomorphic (as Riemann surface) to the unit disc. This deserves some further comments. It was seen that the existence of dicritical singularities is a necessary condition for the leaves of ${{\mathcal F}}, \, {{\widetilde{\mathcal F}}}$ to be non-hyperbolic Riemann surfaces. Indeed a more precise statement holds: in order to have non-hyperbolic leaves, “most” ${\mathcal{H}}$ trajectories induced on the leaves of ${{\mathcal F}}_{\infty}$ must be of finite length. To have finite length implies that the leaf must have both “future and past” ends and, in turn, he existence of “future ends” leads to the existence of the dicritical singularity in our context. In the case of Halphen vector fields, it can be shown by using the ideas of [@guillotIHES] that the trajectories of ${\mathcal{H}}$ are, indeed, finite and that they tile the corresponding leaves of ${{\mathcal F}}$. This might suggest that the leaves of ${{\mathcal F}}$ ought to be “parabolic Riemann surfaces” what is not the case. Explanation for their hyperbolic character is provided [@guillotIHES]) by carefully exploiting the intimate connection between these equations and the Lie algebra of ${\rm PSL}\, (2, {{\mathbb C}})$. Whereas we can hardly improve on [@guillotIHES]), we shall provide in Section 7.2 an explanation of this fact based on the ideas introduced in this paper.

We are now ready to prove Theorem C.

The fact that the singularity produced by a vector field of special type having no dicritical singularity at infinity cannot be realized in a Kähler manifold follows from the combination of Lemma \[Guillotlattice1.1\] with Brunella’s theorem as used in the proof of Lemma \[Cstar1.1\]. In fact, if there were a Kähler manifold equipped with a holomorphic vector field $X$ exhibiting this type of singularity at a point $p$, the blow-up of $X$ at $p$ would endow the exceptional divisor with a foliation whose leaves are hyperbolic Riemann surfaces. Since the remaining leaves associated to the orbits of $X$ must be parabolic (being globally defined $X$ is automatically complete), a contradiction results.

The same argument applies to the case of (hyperbolic) Halphen vector fields since these vector fields are known to have leaves that are isomorphic to the unit disc, cf. [@guillotIHES] or the discussion carried out below.

Consider now the dual singularity of vector field of special type having no dicritical singularity at infinity. This means that we have a neighborhood of $\Delta_{\infty}$ embedded in a manifold $N$. The argument of Theorem \[maintheo\] that an orbit of the vector field entering this neighborhood will contain trajectories converging to the pole locus and such that the integral of the corresponding time-form is finite. This contradicts the completeness of the vector field in the complement of its pole locus. Again the argument for Halphen vector fields is totally analogous.

Poincaré-type series and Halphen vector fields
----------------------------------------------

As a matter of fact, Halphen vector fields constitute a particularly remarkable example of semi-complete vector field belonging to Guillot’s lattice whose geometry and dynamics is nicely described in [@guillotIHES]. As mentioned in this section we are going to discuss some aspects of these vector fields from an alternate point of view. Needless to say that the reader is referred to [@guillotIHES] for a fuller discussion.

In the sequel, we shall work on ${{\mathbb C}}^3$. Let ${\mathcal{E}}$ be the radial vector field ${\mathcal{E}}= x\partial /\partial x + y \partial /\partial y + z
\partial /\partial z$ and set $Y = \partial /\partial x +  \partial /\partial y + \partial /\partial z$. A quadratic homogeneous vector field $X$ is said to be a Halphen vector field if it satisfies $[Y,X] = 2{\mathcal{E}}$. It then follows that the triplet $Y, {\mathcal{E}}, X$ form a Lie algebra isomorphic to the Lie algebra of ${\rm PSL}\, (2, {{\mathbb C}})$. If $Y,X$ and ${\mathcal{E}}$ are identified to the vector fields they induce on $M$ as in Sections 3 and 4 and expressed in the corresponding coordinates $(x,y,z)$, then ${\mathcal{E}}$ becomes $z^{-1} \partial /\partial z$. In other words $X,Y$ commute up to a vertical vector field. Thus $X$ preserves the projection on $\Delta_{\infty}$ of the foliation induced by $Y$. In other words, the foliation ${{\mathcal F}}_{\infty}^X$ induced by $X$ on $\Delta_{\infty}$ is transverse to the foliation induced by $Y$ on $\Delta_{\infty}$. Since $Y$ was a constant vector field, the foliation it induced on $\Delta_{\infty}$ is simply a linear pencil of rational curves. Summarizing ${{\mathcal F}}_{\infty}^X$ is transverse to a linear pencil of rational curves.

Once the above observation is made, it is easy to work out the structure of ${{\mathcal F}}_{\infty}^X$. It leaves exactly $3$ projective lines $C_1, C_2, C_3$ invariant and these $3$ lines intersect mutually at a radial singularity $P \in \Delta_{\infty}$. Indeed the eigenvalues of ${{\mathcal F}}_{\infty}^X$ at $P$ are $1,1$ whereas the eigenvalues of ${{\mathcal F}}^X$ (the foliation induced by $X$ on all of $M$) at $P$ are $1,1,-1$ (the $-1$ eigenvalue being associated to the direction “$z$”). Also, if $P' \in \Delta_0$ is the singularity of ${{\mathcal F}}^X$ “dual” to $p$, then the eigenvalues of ${{\mathcal F}}^X$ at $p'$ are $1,1,1$. For $i=1,2,3$, let $p_i, q_i$ denote the remaining two singularities of ${{\mathcal F}}^X$ over $C_i$. We assume that $X$ is semi-complete though this is not really indispensable in what follows. According to Halphen’s results revisited by Guillot, it easily follows that the eigenvalues of ${{\mathcal F}}_X$ at $p_i$ (resp. $q_i$) and have the form $-1,-1, m_i$ (resp. $-1, -1, -m_i$), with $m_i \in {{\mathbb N}}^{\ast}$. The converse also holds though it is harder to prove, the argument being based on relating the semi-complete character of the vector field to the “uniform” character of its transversely projective structure (for a proof of the converse see [@guillotIHES]). In any event it is also easy to check that ${{\mathcal F}}_X$ is locally linearizable about $p_i$ (resp. $q_i$). Finally note that the convention used above concerning the order of the eigenvalues of ${{\mathcal F}}_X$ at $p_i$ (resp. $q_i$) is such that the first eigenvalue corresponds to the vertical direction “$z$”, the second to the curve $C_i$ and the third to a direction contained in $\Delta_{\infty}$ and transverse to $C_i$. This convention is slightly different from [@guillotIHES] since the eigenvalue associated to the direction “$z$” is denoted by $-1$ rather than $1$. This change of sign is due to the fact that we consider singularities in $\Delta_{\infty}$ whereas Guillot considers singularities in $\Delta_0$.

The dynamics of ${{\mathcal F}}_{\infty}^X$ is fully encoded in its global holonomy group with respect to a fixed line $\overline{C}$ in the above mentioned linear pencil that is transverse to ${{\mathcal F}}_{\infty}^X$. The preceding also shows that this holonomy group coincides with the subgroup of ${\rm PSL}\, (2, {{\mathbb C}})$ generated by three elements $\xi_1, \xi_2, \xi_3$ which are associated to the local holonomy of each of the three invariant lines. In particular we have $$\xi_1 \xi_2 \xi_3 = \xi_1^{m_1} = \xi_2^{m_2}= \xi_3^{m_3} = {\rm id} \, .$$ In other words, it is a triangle group whose dynamics on $S^2$ is well known: provided that $$\frac{1}{m_1} + \frac{1}{m_2} + \frac{1}{m_3}  < 1 \, , \label{stillacrossreference}$$ this group is conjugate to a subgroup of ${\rm PSL}\, (2, {{\mathbb C}})$ and thus it leaves a circle $\Lambda_{\infty} \subset S^2
\simeq \overline{C}$ invariant. Besides each connected component of $\overline{C} \setminus \Lambda_{\infty}$ is invariant by the action. In fact, on these components the action is properly discontinuous whereas it is minimal when restricted to the circle $\Lambda_{\infty}$ itself. Similarly is the case where $1/m_1 + 1/m_2 +1/m_3 =1$, the resulting groups are well-known groups of affine diffeomorphisms associated to special tiles of the plane. When $1/m_1 + 1/m_2 +1/m_3 >1$ the resulting group is indeed finite and thus it is easy to see that all leaves are compact.

[A note of caution in what precedes concerns the fact the quasi-isometric type of the holonomy group does not [*a priori*]{} determine the quasi-isometric type of the leaves of ${{\mathcal F}}_X$ since the latter are not everywhere transverse to the associated fibration: besides the existence of singularities, there are $3$ fibers of this fibration that are invariant under ${{\mathcal F}}_X$. In particular it is not clear that the leaves of ${{\mathcal F}}_X$ must be hyperbolic once Estimate (\[stillacrossreference\]) is verified.]{}

A similar picture is valid for the foliation ${{\mathcal F}}^X$ associated to $X$ on $M$. Clearly ${{\mathcal F}}^X$ is transverse to the codimension $1$ foliation defined by the “cone” over the leaves of ${{\mathcal F}}_{\infty}^Y$ and its dynamics is also encoded in its corresponding holonomy group. This is still generated by the local holonomy [*with respect to $M$*]{} of each of the mentioned three invariant lines. Each of the three generators is now realized as an automorphism $\Xi_i$ of ${\mathbb{F}}_1$, the line bundle over ${{\mathbb C}}P(1)$ with Chern class equal to $1$. In our context this line bundle is the cylinder over $\overline{C} \subset \Delta_{\infty}$. Besides ${\mathbb{F}}_1$ can be obtained by gluing together two copies of ${{\mathbb C}}\times {{\mathbb C}}$ with coordinates $(w,z)$ and $(w', z')$ according to the equation $(w', z') = (1/w , wz)$. The coordinate “$z$” of the first copy can be identified with the previous affine coordinate “$z$” for ${{\mathbb C}}^3$. The automorphism $\Xi_i$ fixes the null-section and thus it can be expressed in the mentioned coordinates as $$\Xi_i (w,z) = (\xi_i (w) , B_i(w) z)$$ where $\xi_i (w)$ is a homography. Furthermore it is also know that $\Xi_i^{m_i} = {\rm id}$ on ${\mathbb{F}}_1$ and $\xi_i^{m_i} = {\rm id}$ on $\overline{C}$. Thus $$B_i(w) \times B_i( \xi_i (w)) \times \cdots \times B_i (\xi_i^{m_i-1} (w)) = 1 \, . \label{product=1}$$

Next recall that the Möbius group has a natural extended action to ${\mathbb{F}}_1$ consisting of multiplying vectors in the fibers by the square root of its derivative. In other words, if $\xi$ is a homography, then its extended action on a pair $(w,z) \in {\mathbb{F}}_1$ is simply $$\xi . (w,z) = (\xi (w) , \sqrt{\xi' (w)} \, z ) \, .$$ It is to be noted that the square root of the derivative of a homography is well-defined so that the claim follows from observing that ${\mathbb{F}}_1 \otimes {\mathbb{F}}_1$ is isomorphic to the tangent bundle of $\overline{C}$. In particular a more explicitly expression for $\Xi_i$ can be derived as follows. Let $q_i, q_{i+1}$ ($\neq 0, \infty$) denote the two fixed points of $\xi_i$ in $\overline{C}$. The transformation $$\sigma_i (w) = \frac{w-q_i}{w-q_{i+1}} \; \; \; \;  ; \; \; \; \; \sigma_i^{-1} (w) =\frac{q_{i+1} w -q_i}{w-1}$$ conjugates $\xi_i$ to a homography fixing $0, \infty$. In this coordinate $\overline{w}$, $\xi_i$ must take on the form $\overline{w} \mapsto k_i^2 \overline{w}$ where $k_i^2 = e^{2\pi \sqrt{-1}/m_i}$ since $\xi_i^{m_i} = {\rm id}$ on $\overline{C}$. Furthermore, in the coordinate $\overline{w}$ it is clear that $B_i (\overline{w})$ must be constant so as to allow $\Xi_i$ to have a holomorphic extension to a fibered neighborhood of $\infty$. Setting $B_i (\overline{w}) =B_i$ for this particular choice of coordinates, it follows that the expression of $\Xi_i$ in the initial coordinate $w$ is given by $$\begin{aligned}
\Xi_i (w,z) &=& (\xi_i (w) , B_i (w) z) = \left( \sigma_i^{-1} (k_i^2 \sigma_i (w)) , \sqrt{(\sigma_i^{-1})'\vert_{k_i^2 \sigma_i (w)}} B_i \sqrt{\sigma_i' (w)} z \right) \\
& = & \left( \frac{(k_i^{-1} q_i -k_i q_{i+1}) w + q_i q_{i+1} (k_i -k_i^{-1})}{(k_i^{-1} -k_i) w + (k_i q_i -k_i^{-1} q_{i+1})} ,
\frac{k_i^{-1} B_i (q_i -q_{i+1}) z}{(k_i^{-1} -k_i) w + (k_i q_i -k_i^{-1} q_{i+1})} \right) \\
& = & (\xi_i (w) , k_i^{-1} B_i \sqrt{\xi_i' (w)} z ) \, .\end{aligned}$$ The above formulas are going to enable us to understand the solutions of Halphen vector fields from the point of view worked out in this work. Let us first consider the special case where $1/m_1 + 1/m_2 +1/m_3 =1$. In this case the three homographies $\xi_1 ,\xi_2 , \xi_3$ share a common fixed point. Choosing coordinates $(w,z)$ where this point is $\infty$ it follows that $\xi'(w)$ is constant and thus $\Xi_i (w,z) = (\xi_i (w) , A_i z)$ for certain constants $A_i$, $i=1,2, 3$. As a sort of converse to Theorem \[maintheo\], we obtain the following:

\[entireHalphen\] Under the above conditions all the leaves of ${{\mathcal F}}^X$ are uniformized by ${{\mathbb C}}$ as Riemann surfaces.

The proof is rather simple. Let $L$ be a generic leaf of ${{\mathcal F}}_X$ and let ${\mathbb{F}}_1$ be as above. Because all the $A_i$’s are constant, the intersection points $p_1, p_2 \ldots$ between $L$ and ${\mathbb{F}}_1$ have their distance to $\Delta_{\infty}$ bounded from below by a positive constant. Thus the time-form $dT_L$ induced by $X$ on $L$ is uniformly bounded on a neighborhood $W$ of $\{ p_1, p_2 \ldots \}$. Consider now the maximal domain ${\bf U} \subseteq {{\mathbb C}}$ of a solution $\phi$ of $X$. Suppose for a contradiction that ${{\mathbb C}}\setminus {\bf U} \neq \emptyset$ and choose a point $T \in {{\mathbb C}}$ lying in the boundary $\partial {\bf U}$ of ${\bf U}$. Finally let $t_1, t_2, \ldots $ be a sequence of points in ${\bf U}$ converging to $T$. Given the above mentioned structure of ${{\mathcal F}}_X$ as a foliation transverse to a fibration, we can assume without loss of generality that $\phi (t_j)$ lies in $W$. Since $dT_L$ is uniformly bounded on $W$, it follows that $\phi$ is defined on a disc of uniform (positive) radius about each $t_j$. This is however impossible since $t_j \rightarrow T \in \partial {\bf U}$. The proposition is proved.

From now on let us focus on the more interesting case where $\xi_1, \xi_2, \xi_3$ generate a hyperbolic triangle group. The fixed points of $\xi_1, \xi_2, \xi_3$ are three (mutually different) points $q_1 ,q_2, q_3 \in \overline{C}$. By this we mean that for each $i \in \{ 1,2,3 \}$, the homography $\xi_i$ fixes the points $q_i, q_{i+1}$ (where $q_{3+1} =q_1$). In the present case there is no coordinate $w$ where all the $\xi_i$ become affine maps. Thus we shall need to work with the full information provided by the action of $\Xi_1, \Xi_2, \Xi_3$ on ${\mathbb{F}}_1$. We are going to show that the geometry of the leaves is related to the Poincaré series with exponent $1/2$. Next let us denote by $\Gamma$ the group generated by $\xi_1, \xi_2, \xi_3$ and consider its Cayley graph with respect to the generating set given by $\xi_1, \xi_2, \xi_3$ and their inverses. Choose a “geodesic ray” $\gamma_0 ={\rm id}, \gamma_1, \gamma_2 , \ldots$ in the Cayley graph going from the identity to an “end” of the graph. We have:

\[hyperbolic-vs-entire\] Let $L$ denote a leaf of ${{\mathcal F}}_X$ passing through a point $(w_0, z_0)$. As a Riemann surface $L$ is hyperbolic provided that the series $$S (w_0) = \sum_{j=0}^{\infty} \Vert \gamma_j (w_0) \Vert^{1/2} \label{georay}$$ is convergent for more than one geodesic ray. If this series diverges for all geodesic rays as above then $L$ is a quotient of ${{\mathbb C}}$.

The proof consists of elaborating further on the argument used in Proposition \[entireHalphen\]. Again denote by ${\bf U}$ the maximal domain of definition of the solution $\phi$ of $X$ satisfying $\phi (0) =(w_0, z_0)$, $z_0 \neq 0$. Denote by $L_{\infty}$ the projection of $L$ on $\Delta_{\infty}$. By virtue of the structure of the foliation ${{\mathcal F}}_{\infty}^X$ on $\Delta_{\infty}$, we know that $L_{\infty}$ is a ramified covering of ${{\mathbb C}}P(1)$ where the ramified points sit over three points of ${{\mathbb C}}P(1)$ (identified to the three invariant fibers of ${{\mathcal F}}_{\infty}^X$). We can then think of $L_{\infty}$ as being the universal covering of ${{\mathbb C \mathbb P}}(1)$ minus $3$ points modulo adding to it the ramification points. In particular there is a natural sense in considering [*fundamental domains*]{} in $L_{\infty}$. The leaf $L_{\infty}$ can then be considered as the union of the corresponding fundamental domains $L_{\infty}^{(0)},  L_{\infty}^{(1)}, \ldots$ such that $L_{\infty}^{(j)} = \gamma_j (L_{\infty}^{(0)})$. These domains have natural lifts to the leaf $L \subset M\setminus (\Delta_0 \cup \Delta_{\infty})$ which, modulo re-numeration, will be denoted by $L^{(0)},  L^{(1)}, \ldots$ in such way that $L^{(j)} = \Gamma_j (L^{(0)})$ where $\Gamma_j$ is the automorphism of ${\mathbb{F}}_1$ corresponding to the action of $\gamma_j$ on $\overline{C}$.

On $L$ (or on its universal covering if necessary), we define the map $$\mathcal{D}_L (p) = \int_{(w_0, z_0)}^p dT_L$$ where $dT_L$ stands for the time-form induced on $L$. Since $X$ is semi-complete, $\mathcal{D}_L$ provides a diffeomorphism from the (universal covering of) $L$ to ${\bf U}$. Let then $U_{(j)}$ be the image of $L^{(j)}$ by $\mathcal{D}_L$. The above assertion implies that the set $U_{(j)}$ “tile” ${\bf U}$ without overlapping and modulo adding the image of ramification points (involved in the preceding definition of the fundamental domains $L_{\infty}^{(0)},  L_{\infty}^{(1)}, \ldots$). Next recall that the affine structure on $L_{\infty}$ is uniformly bounded (from below and by above). Combining this fact to the expression for $dT_L$ arising from Formula \[tXX\], with $d=2$, it follows the existence $0<r_j < R_j$, $j =1, 2 ,\ldots$, satisfying the conditions below.

1.  There are constants $0<c<C$, independent of $j$, such that $$c \Vert \pi_2 (\Xi_j (w_0, z_0)) \Vert  r_j < R_j < C \Vert \pi_2 (\Xi_j (w_0, z_0)) \Vert  r_j \,   ,$$ where $\pi_2$ stands for the projection on the second coordinate (i.e. the fiber of ${\mathbb{F}}_1$).

2.  The image of $U_{(j)}$ under $\mathcal{D}_L$ contains a ball of radius $r_j$ about $\mathcal{D}_L (\Xi_j (w_0, z_0))$. Similarly the same image is contained in ball of radius $R_j$ about $\mathcal{D}_L (\Xi_j (w_0, z_0))$.

It then becomes clear that ${\bf U}$ must be the whole ${{\mathbb C}}$ provided that the series $\sum_{j=0}^{\infty} \Vert \pi_2 (\Xi_j (w_0, z_0) \Vert$ diverges for every geodesic ray. Conversely, if this series is convergent, then we can easily construct a “small” piece of continuum contained in the boundary of ${\bf U} \subset {{\mathbb C}}$. Thus ${\bf U}$ must be a hyperbolic domain so that $L$ itself must be a Riemann surface covered by the unity disk. To conclude the proof of the proposition it is therefore sufficient to check that the series $\sum_{j=0}^{\infty} \Vert \pi_2 (\Xi_j (w_0, z_0) \Vert$ converges (resp. diverges) if and only if so does the “reduced Poincaré series” in the statement. This is however clear since $\vert k_i^{-1} B_i \vert =1$ as an immediate consequence of Formula (\[product=1\]). The proof of the proposition is over.

Next we are going to show that the series (\[georay\]) always converges provided that $w_0 \in \overline{C} \setminus
\Lambda_{\infty}$. Some indications concerning the behavior of this series in the case $w_0 \in \Lambda_{\infty}$ will also be provided. As mentioned the series in question differs from the usual Poincaré series since the sum is not carried over the entire group but only over those elements belonging to a chosen “geodesic ray”. Indeed the “full” Poincaré series of $\Gamma$ with exponent $1/2$ diverges as it follows from well-known results due mainly to Sullivan (see [@bulletin] for an overview of the standard theory).

\[alwaysconverging\] The series (\[georay\]) converges provided that $w_0 \in \overline{C} \setminus \Lambda_{\infty}$.

The argument is rather simple. Consider the action of $\Gamma$ in the $w$-plane. This action preserves a circle identified to $\Lambda_{\infty}$. In particular $\Gamma$ is realized as a Fuchsian group, i.e. a discrete group of automorphisms of the hyperbolic disc. Next, since $\Gamma$ acts on the hyperbolic disc, it is easy to see that the convergence of this series does not depend on $w_0$. In other words, the series (\[georay\]) converges for $w_0$ if and only if it converges at $0$. It is then sufficient to check that the series converges for $w_0 =0$ (identified to the origin of the disc). For this consider again the geodesic ray $\gamma_0 ={\rm id}, \gamma_1, \gamma_2 , \ldots$ in the Cayley graph of $\Gamma$ and set $a_j = \gamma_j (0)$. In the mentioned Cayley graph the distance between the identity and $\gamma_j$ is obviously $j$. The existence of a quasi-isometry between this graph and the hyperbolic disc implies that the hyperbolic distance $d_H (0, \gamma_j (0))$ between $0$ and $\gamma_j (0)$ satisfies $$c j < d_H (0, \gamma_j (0)) < C j$$ for appropriate uniform constants $C > c >0$. The standard formula for the length of a minimizing geodesic in the hyperbolic unit disc joining $0$ to a point $a$ of this disc (naturally satisfying $\Vert a \Vert <1$) yields $$\frac{e^{cj} -1}{e^{cj} + 1} \leq \Vert a_j \Vert \leq \frac{e^{Cj} -1}{e^{Cj} + 1} \, .$$ On the other hand the coefficients of the hyperbolic metric at $0$ and at $a_j$ allow us to obtain a formula for the derivative of $\gamma_j$ at $0$. Combined with the above estimates this formula gives $$\Vert \gamma_j' (0) \Vert \leq \frac{4e^{Cj}}{(1+ e^{Cj})^2}$$ and thus $\Vert \gamma_j' (0) \Vert^{1/2} \leq 2 e^{Cj/2} / (e^{Cj} +1)$. The convergence of the mentioned series follows immediately.

To close this discussion let us briefly indicate the behavior of the series (\[georay\]) for points $w_0$ lying in $\Lambda_{\infty}$. Since $\gamma_j$ takes $0$ to $a_j$, it follows that $\gamma_j (w) = e^{2\pi i \theta} (w+a_j)/
(1 + \overline{a}_j w)$, for some $\theta \in [0,1)$. In particular $$\Vert \gamma_j' (w) \Vert = \frac{1 - \Vert a_j \Vert^2}{(1 + \overline{a}_j w)^2} \, .$$ Because $\Gamma$ is discrete, we know that $\Vert a_j \Vert \rightarrow 1$ when $j \rightarrow + \infty$. Next set $a_j = \Vert a_j \Vert e^{i\theta_j}$ and $w_0 = e^{-i\theta_j -\pi + \alpha_j}$ so that $$\Vert \gamma_j' (w_0) \Vert = \frac{1 - \Vert a_j \Vert^2}{1 + \Vert a_j \Vert^2 -2\Vert a_j \Vert \cos (\alpha_j)} \, . \label{justtobequoted}$$ Next note that $\Vert \gamma_j' (w_0) \Vert > 1/2$ as long as $\cos (\alpha_j) > \Vert a_j \Vert$. In particular if there are infinitely many indices $j$ satisfying this condition that the corresponding series will diverge. Clearly the existence of infinitely many indices as indicated means that some subsequence of the $a_j$ converge “almost radially” for $-w_0$. Conversely if the denominator in (\[justtobequoted\]) is bounded from below by some positive constant, then the argument used in Lemma \[alwaysconverging\] ensures again the convergence of series (\[georay\]). In general we are led to a finer analysis taking into account the “conic approximation” of $-z_0$ by the sequence $a_j =\gamma_j (0)$. Not surprisingly the behavior of series (\[georay\]) on $\Lambda_{\infty}$ depends on the initial point $w_0$: for some values of $w_0$ it diverges whereas for others it is convergent.

Most properties of Halphen vector fields become encoded in the extended dynamics of the group generated by $\Xi_1, \Xi_2, \Xi_3$ on ${\mathbb{F}}_1$. For example the study of first integrals for Halphen vector fields amounts to searching functions that are invariant by this action. In particular on each connected component of $\overline{C} \setminus \Lambda_{\infty}$ it is not hard to construct “automorphic functions” for this group so that on the corresponding open sets on $M$ the Halphen vector field possesses a holomorphic first integral. It is also not very hard to check that the Halphen vector field does not admit a holomorphic (or meromorphic) first integral on the set corresponding to $\Lambda_{\infty}$ (which has real dimension equal to $5$). Yet on the latter set, there is a real-valued first integral for the equation that is actually globally defined on $M$. We shall not pursue this type of discussion here not only because Halphen vector fields were detailed studied in [@guillotIHES] but also because the corresponding issues will be no longer in line with the main ideas of this paper.

However, inasmuch none of the above results about Halphen vector fields is new, our method is applicable in other situations since, in most of our discussion, we have only needed the fact that Halphen vector fields are transverse to a singular rational fibration. Since many remarkable equations, such as Painlevé equations, admit a formulation in terms of rational vector fields transverse to suitable singular foliations, it is very reasonable to expect that similar arguments can provide new insight in other situations of interest an example of it being provided by the material in Section 7.3.

It also interesting to consider differential equations whose solutions are meromorphic functions (defined on ${{\mathbb C}}$). This is a very classical topic not only in differential equations and integrable systems but also in Nevanlinna’s theory. The methods presented in the course of this work seem well-designed to investigate some of these issues. In fact, for example, Painlevé equations equations may be expressed as a rational vector field on ${{\mathbb C}}^3$. More generally every non-autonomous second-order differential equations is equivalent to a rational vector field on ${{\mathbb C}}^3$ which, actually, has a very special nature. Whereas these vector fields may be rational rather than “polynomial”, this is of relatively little importance for our methods. Essentially the pole divisor may be displaced from the “plane at infinity” to some other algebraic surface and our methods will still be effective in this case.

In the case where we have vector fields as above having solutions that are [*meromorphic functions defined on ${{\mathbb C}}$*]{}. Consider also the foliation ${{\mathcal F}}_{\infty}$ induced on the “plane at infinity” by this vector field. Modulo very mild assumptions concerning the singularities of the corresponding vector fields at infinity, the methods of Theorem \[maintheo\] and of Proposition \[hyperbolicleaves\] will imply that all the ${\mathcal{H}}$-trajectories defined on ${{\mathcal F}}$ must be of finite length (in particular they must have future-end singularities). This seems to be a very restrictive condition on the foliation ${{\mathcal F}}_{\infty}$ and it is therefore conceivable to expect this issue can further be exploited to shed some new light in these very classical problems.

In general foliations for which all ${\mathcal{H}}$-trajectories are of finite length seem very special. In this direction we would like to propose:

[**Conjecture**]{}: Foliations all of whose ${\mathcal{H}}$-trajectories are of finite length are always transverse to a singular fibration (possibly modulo passing to some abelian covering).

If this conjecture holds, then the ideas developed here can be applied to every rational vector field since either we will have ${\mathcal{H}}$-trajectories with infinite length to “produce” hyperbolic domains or we shall have a type of Poincaré series for the corresponding global holonomy group to be studied.

Singular points of holomorphic vector fields on compact Kähler threefolds
-------------------------------------------------------------------------

As promised this last section is devoted to describing a procedure leading to the solution of the following problem:

[*Problem*]{}: Let $X$ be a globally defined holomorphic vector field on a compact $3$-dimensional Kähler manifold $N$. Suppose that $p \in N$ is such that $X(p) = 0$. Find all possible normal forms for the first non-zero homogeneous component $X^h$ of $X$ at $p$.

Whereas we shall not work out the desired list of normal forms here, we are going to give a detailed description of a procedure eventually leading to it. Naturally to fully implement this procedure is a reasonably elaborated task that would take quite a few pages so that there is no point in trying to carry it out in this paper. On the other hand, let us point out that the same “list” of normal forms also contains all the possibilities for the top degree homogeneous component of a complete polynomial vector field on ${{\mathbb C}}^3$. Indeed all the arguments presented in the subsequent discussion can immediately be transferred to the case of complete polynomial vector fields by using the methods already explained in the previous sections.

To begin with by means of a local coordinate we identify $X$ to a holomorphic vector field defined on a neighborhood of $(0,0,0) \in {{\mathbb C}}^3$. In particular $X^h$ is simply the first non-zero homogeneous component of the Taylor series of $X$ at the origin so that it is defined on all of ${{\mathbb C}}^3$. Besides, we can assume that $X^h$ is not a multiple of the radial vector field since otherwise there is nothing to be proved. In particular $X^h$ induces a vector field. This allows us to consider $X^h$ as being defined in the manifold $M$ obtained by blowing-up ${{\mathbb C}}P(3)$ at the “origin” exactly as previously done in this work. In particular $X^h$ induces a foliation ${{\mathcal F}}^h$ on $M$ and ${{\mathcal F}}^h$ leaves invariant the planes $\Delta_0$ and $\Delta_{\infty}$. As already seen, the fact that $X$ is a globally defined vector field on a compact Kähler manifold implies the following:

$\bullet$ $X^h$ is semi-complete.

$\bullet$ The leaves of ${{\mathcal F}}^h$ are not hyperbolic Riemann surfaces. In particular the foliations ${{\mathcal F}}^h_0, \, {{\mathcal F}}^h_{\infty}$, obtained by restriction of ${{\mathcal F}}^h$ to $\Delta_0, \, \Delta_{\infty}$, have no hyperbolic leaf.

We can also assume that the foliation ${{\mathcal F}}^h$ does not coincide with a pencil of rational or of elliptic curves for otherwise normal forms for $X^h$ can easily be obtained. Indeed recall that rational pencils on $\Delta_0$ were classified long ago by Suzuki cf. [@suzuki]. Although the analogous question about elliptic pencils is a lot more complicated, the fact that $X^h$ is semi-complete makes the question much more treatable. So we shall look for the cases where the foliations ${{\mathcal F}}^h_0, \, {{\mathcal F}}^h_{\infty}$ do not coincide with a pencil. These foliations are clearly identified through the construction of Section $3$ and, as usual, it suffices to work with ${{\mathcal F}}^h_{\infty}$. Since ${{\mathcal F}}^h_{\infty}$ is defined on the $2$-dimensional manifold $\Delta_{\infty} \simeq {{\mathbb C}}P(2)$, a deep result by McQuillan implies that they are transverse to a rational or to an elliptic pencil defined on $\Delta_{\infty}$, cf. [@book]. The result of McQuillan itself, whereas very powerful, falls short of solving our question. in fact, our problem is in principle harder than the problem of classifying (non-homogeneous) polynomial vector fields on ${{\mathbb C}}^2$ which has itself required non-trivial elaboration on McQuillan techniques cf. [@marco3], [@bruno] (see also [@guillotreb] for a more general result dispensing with McQuillan’s theorem).

In any event we shall combine McQuillan’s theorem with the ideas developed in Section 7.2 to work out normal forms for $X^h$. This goes as follows. As in the previous paragraph, the global dynamics of ${{\mathcal F}}^h_{\infty}$ is encoded in its (global) holonomy group which is itself a subgroup of the autormorphism group of a curve $\overline{C}$. The curve $\overline{C}$ is either rational or elliptic. To abridge the discussion let us consider only the case in which $\overline{C}$ is a rational curve. The global holonomy group $\Gamma$ is then a subgroup of ${\rm PSL}\, (2, {{\mathbb C}})$. Next we have:

\[finishing1\] The holonomy group $\Gamma$ is conjugate to a subgroup of ${\rm Aff}\, ({{\mathbb C}})$.

Suppose for a contradiction the statement is false. It follows that $\Gamma$ is not solvable. Therefore either $\Gamma$ is discrete or it is dense in ${\rm PSL}\, (2, {{\mathbb C}})$. If $\Gamma$ is discrete then the estimates carried out in the proof Lemma \[alwaysconverging\] can be repeated to obtain a contradiction with the fact the leaves of ${{\mathcal F}}^h$ are not hyperbolic Riemann surfaces. Similarly, if $\Gamma$ is dense in ${\rm PSL}\, (2, {{\mathbb C}})$, then $\Gamma$ contains an infinite discrete subgroup $\Gamma_0$. The ends of the Cayley graph of $\Gamma_0$ corresponds to ends of the leaves of ${{\mathcal F}}^h$. Thus once again the estimates of Lemma \[alwaysconverging\] applied to $\Gamma_0$ leads to a contradiction. The lemma is proved.

Lemma \[finishing1\] allows us to suppose that $\Gamma \subseteq {\rm Aff}\, ({{\mathbb C}})$. Resuming the notations of the preceding paragraph, we choose generators $\Xi_1, \ldots , \Xi_r$ for $\Gamma$. Again we set $\Xi_i (w,z) = (\xi_i (z) , A_i z)$ where the coordinates $(w,z)$ are chosen so that the common fixed point of the homographies $\xi_i$ is identified to $\infty$. In particular each $A_i$ is constant in the sense that it does not depend on the variable $z$. Now the argument used in Proposition \[entireHalphen\] combined to the fact the leaves of ${{\mathcal F}}^h$ are parabolic Riemann surfaces immediately yields:

\[finishing2\] For every $i \in \{ 1, \ldots ,r\}$, we have $\Vert A_i \Vert =1$.

It follows from Lemma \[finishing1\] that ${{\mathcal F}}^h$ possesses an algebraic curve $\overline{B}$ that is not a fiber of the rational fibration in question. Indeed $\overline{B}$ corresponds to the fixed point of $\Gamma$ on $\overline{C}$. We shall now use some of the techniques introduced in [@guillotreb]. The semi-complete vector field $X^h$ induces a [*uniformizable*]{} affine structure on $\overline{B}$. In particular $\overline{B}$ must be either rational or elliptic. Furthermore the corresponding uniformizable affine structures were also classified in the same paper. When $\overline{B}$ is an elliptic curve the affine structure is either the standard “flat” structure or the so-called Hopfian torus. The case of Hopfian torus can however be ruled out since it would imply that some $A_i$ has modulos different from $1$, thus contradicting Lemma \[finishing2\]. The other possibility corresponds to having $\overline{B}$ as a rational curve. The number of possibilities for the affine structure in question is slightly larger. With the notations of [@guillotreb], we have:

1.  Rational orbifold of type $n >0$.

2.  The parabolic cylinder.

3.  The orbifold $(2,2, \infty)$. [@guillotreb]

4.  One of the elliptic orbifolds $(2,3,6)$, $(2,4,4)$, $(3,3,3)$, $(2,2,2,2)$.

It also clear that the singularities of the affine structure on $\overline{B}$ corresponds to invariant fibers of the rational pencil. The converse is less clear. In fact, the crucial step to be able to carry out the classification of the possible homogeneous vector field $X^h$ is as follows.

\[finishing3\] Suppose that $p \in \overline{B}$ is an intersection point of $\overline{B}$ with an invariant fiber of the fibration in question. Then the affine structure induced by $X^h$ on $\overline{B}$ cannot be regular at $p$.

If we take for grant the statement above, the classification of $X^h$ can be concluded as follows:

-   $\overline{B}$ must be a rational curve. Indeed, over an elliptic curve the uniformizable affine structure have no singular point. Thus the fibration transverse to ${{\mathcal F}}^h$ has no invariant fiber. Therefore ${{\mathcal F}}^h$ must be transverse to the mentioned fibration everywhere and this is impossible since ${{\mathcal F}}^h$ has singular points.

-   From now on we consider that $\overline{B}$ is a rational curve. Let us first suppose that the affine structure induced on $\overline{B}$ by $X^h$ is the parabolic cylinder. In this case it becomes clear that ${{\mathcal F}}^h$ is induced by a linear vector field so that nothing else need to be said about this case.

-   Next we recall a general statement of [@guillotreb], namely the so-called “Fundamental Lemma” asserting in particular the following: suppose that $p \in \overline{B}$ is a singular point for the affine structure on $\overline{B}$. Then the local holonomy of $\overline{B}$ about $p$ has order determined by the ramification index of the affine structure at $p$. This order is finite unless there is no ramification.

-   Complementing the previous comment, let us point out that the singular points of the above described uniformizable affine structures where ramification vanishes are precisely the two singular points of the “parabolic cylinder” structure and the point indicated as $\infty$ in the orbifold $(2,2,\infty)$.

-   We now return to the list of possible affine structures. The case of the “rational orbifold of type $n >0$” can immediately be ruled out. The local holonomy maps are finite and they commute. Thus the holonomy group $\Gamma$ is finite and therefore this affine structure can only be produced if ${{\mathcal F}}^h$ is itself a pencil.

-   The elliptic orbifolds $(2,3,6)$, $(2,4,4)$, $(3,3,3)$. Here we immediately conclude that $\Gamma$ must be a discrete subgroup of ${\rm Aff}\, ({{\mathbb C}})$. These cases therefore correspond to the three “special” Halphen vector fields singled out in Proposition \[entireHalphen\].

-   Finally it remains to consider the cases of the orbifolds $(2,2, \infty)$ and $(2,2,2,2)$. These two groups have similar representations (with infinite image) on ${\rm Aff}\, ({{\mathbb C}})$. The resulting “Riccati equation” is therefore uniquely determined. It would only remains to check whether or not is comes from a semi-complete homogeneous polynomial vector field. This is probably not the case.

To finish this discussion let us indicate how we can try to establish the statement \[finishing3\] or, if this statement is false, the understand the structure of invariant fibers giving rise to regular points for the affine structure of $\overline{B}$. The idea to approach this problem relies on adapting the “combinatorics” developed in Section 6 of [@guillotreb] to the current case. For this adaptation to make sense the key remark is the “reduction theorem” proved in the Appendix of [@julioindiana]. In fact, according to these results the singularities of the vector field $X^h$ (defined on a $3$-dimensional ambient) can be reduced through usual “cylindric” blow-ups. With this reduction performed, let us consider a singularity $q$ of the corresponding reduced foliation $\widetilde{{{\mathcal F}}}^h$. The singularity $q$ belongs to an invariant plane for $\widetilde{{{\mathcal F}}}^h$ and, locally, we can find coordinates $(u,v,w)$ about $q$ where $\widetilde{{{\mathcal F}}}^h$ coincides with the local orbits of a vector field has the form $$a(u,v) \partial /\partial u + b(u,v) \partial /\partial v + w c(u,v) \partial /\partial w \, ,$$ where $q \simeq (0,0,0)$. Besides the singularity of the $2$-dimensional vector field $a(u,v) \partial /\partial u + b(u,v) \partial /\partial v$ at $(0,0) \in {{\mathbb C}}^2$ is redued in the usual Seidenberg sense. All this follows from the reduction procedure presented in [@julioindiana].

Next observe that, if $a(u,v) \partial /\partial u + b(u,v) \partial /\partial v$ has two eigenvalues different from zero at $(0,0) \in {{\mathbb C}}^2$ then it possesses two separatrices $S_1, S_2$ (which in the $3$-dimensional ambient are contained in the plane $\{ w=0\}$). Furthemore the ramification indices of the affine structures induced in these separatrices can be put in natural correspondence. This correspondence can be viewed as a version of the [*reciprocity relation*]{} in Section 6 of [@guillotreb] (Equation (15)) and it is enough to start up the combinatoric study of the corresponding arrangement of invariant curves.

A new difficulty appearing in our case corresponds to the case in which $a(u,v) \partial /\partial u + b(u,v) \partial /\partial v$ possesses exactly one eigenvalue equal to zero at $(0,0) \in {{\mathbb C}}^2$, i.e. the corresponding $2$-dimensional singularity is a saddle-node. In this case the reciprocity relation does not have an immediate generalization and this might create some new difficulties. However the detailed understanding of these singularities obtained in Section 3 of [@julioindiana] is likely to help us to fill in all the gaps.

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[Julio Rebelo]{}\
Institut de Mathématiques de Toulouse\
118 Route de Narbonne\
F-31062 Toulouse, FRANCE.\
rebelo@math.univ-toulouse.fr

[Helena Reis]{}\
Centro de Matemática da Universidade do Porto,\
Faculdade de Economia da Universidade do Porto,\
Portugal\
hreis@fep.up.pt\
