\K -Einstein metrics and compactifications of ${\Bbb{C}}^n$
\hspace{1cm}Universit\`{a} di Parma--Italy \hspace{2cm} Universit\`{a}\ di Sassari--Italy
math0103119.tex
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\newtheorem{teor}{Theorem}[section]
\newtheorem{defin}[teor]{Definition}
\newtheorem{remar}[teor]{Remark}
\newtheorem{prop}[teor]{Proposition}
\newtheorem{corol}[teor]{Corollary}
\newtheorem{lemma}[teor]{Lemma}
\newtheorem{guess}{Conjecture}
\newtheorem{problem}{Problem}
\newtheorem{Example}[teor]{Example} 


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\title{\K -Einstein metrics and
compactifications of ${\Bbb{C}}^n$}
\author{Claudio Arezzo\thanks{e-mail: claudio.arezzo@unipr.it}
\hspace{4cm}
Andrea Loi\thanks{e-mail: loi@ssmain.uniss.it}\\
\hspace{1cm}Universit\`{a} di Parma--Italy \hspace{2cm}
Universit\`{a}\ di Sassari--Italy}

\date{} \begin{document} \maketitle

{\small{{\it{Keywords}}: \K\ metrics; \K\--Einstein metrics;
compactifications of ${\Bbb{C}}^n$; diastasis; Bochner's coordinates.

{\it{Subj.Class}}: 32L07, 53C25, 53C55.}}




\section{Introduction and statements of the main results}\label{Introduction}
The study
of holomorphic 
and isometric immersions
of a
\K\
manifold
$(M, g)$
into a \K\
manifold
$(N, G)$
started with 
Calabi.
In his famous paper
\cite{ca}
he considered the case
when the ambient
space $(N, G)$ is a complex
space form, i.e. its
holomorphic
sectional curvature 
$K_N$ is constant.

There are three types of complex space
forms : \emph{flat, hyperbolic or
elliptic} according as the holomorphic sectional
curvature is \emph{zero, negative, or positive}.
Every  $n$-dimensional
complex space form
is, after multiplying
by a suitable constant, locally
holomorphically isometric to one of the 
following:
the complex euclidean space ${\complex}^{n}$
with the flat metric;
the unit ball in ${\complex}^{n}$
with its Bergman metric
of negative holomorphic
sectional curvature;
the $n$-dimensional complex
projective space ${\complex}P^{n}$ 
endowed with the
Fubini-Study metric 
$g_{FS}$ of positive holomorphic
sectional curvature. 
Recall that $g_{FS}$
is the metric whose
associated 
\K\ form is given by
\begin{equation}\label{kalerproj}
\omega_{FS}=\frac{i}{2}
\partial\bar{\partial}\log\displaystyle
\sum _{j=0}^{n}|Z_{j}|^{2}
\end{equation}
for a homogeneous coordinate
system
$[Z_0,\dots , Z_{n}]$.


One of Calabi's main results
asserts that if a complex
space form can be holomorphically
and isometrically
immersed into another
complex space form then their
sectional holomorphic curvatures
have the same sign.

The natural and interesting extension
of this theorem is the case 
when the metric $g$ is \K\--Einstein
(K--E from now on)
and $(N, G)$ is 
a complex space form.
The first result in this 
direction is due to Chern
\cite{ch} who showed that
if $M$ is
a hypersurface
immersed
in $N$ then 
$M$ is
totally geodesic
if $K_N\leq 0$, while if $K_N>0$
then 
either $M$ is totally geodesic
or a quadric.

These results where further
generalized by Umehara
\cite{um} 
and Tsukada \cite{ts}.
The first
proved that, if
$K_N\leq 0$, 
and $(M, g)$ is
K--E 
manifold holomorphically and
isometrically immersed
in $N$ then 
it is totally geodesic.
The second
showed that,
if $K_N>0$,
and the codimension
of $M$ in $N$
is $2$, then 
$M$ is totally geodesics
or a quadric. A systematic study for general
codimension, in the
case $K_N>0$,
is due to Hulin
in \cite{hu} and \cite{hu1}. 
Among other
things she showed that 
if $(M, g)$ is a
K--E embedded
submanifold of
$({\complex}P^n, g_{FS})$
then its scalar 
curvature is
strictly positive.





The aim of this paper
is to extend Hulin's proof
so to generalize her
result
to embedded 
compact
K--E submanifolds
of projectively 
algebraic
\K\ manifolds
of $(N, G)$
which share same
holomorphic and isometric
properties
with  
$({\complex}P^n, g_{FS})$.
More precisely 
we assume throughout
this paper
that $N$
contains an analytic
subvariety $Y$, not necessarily smooth,
such that 
$X=N\setminus Y$
is 
biholomorphic
to ${\complex}^n$.
These spaces 
are called
{\em compactifications
of ${\complex}^n$}
and they
have been extensively
studied being the object of
a famous problem
posed by Hirzebruck
\cite{hi}. Of course ${\complex}P^n$ is 
an example of these spaces, since $Y={\complex}P^{n-1}$
satisfies all the required properties.

As far as the \K\ metric $G$ is concerned, we
impose three conditions
which are clearly satisfied
by the Fubini--Study metric
$g_{FS}$ on ${\complex}P^n$
(see Section \ref{condA}).
\newline
{\em Condition 1: $G$ is real analytic.}
This condition,
which is clearly very natural
in this setting,
has been extensively studied
by Calabi \cite{ca}.
The ingenious idea of 
Calabi was the introduction,
in a neighborhood of a point 
$p\in N$
of a very special
\K\ potential
$D_p$ for the metric
$G$, which 
he christened 
{\em diastasis}.
Recall that
a \K\ potential
is a
analytic function 
$\Phi$ 
defined in a neighborhoood
of a point $p$
such that 
$\Omega =\frac{i}{2}\bar\partial\partial\Phi$,
where $\Omega$
is the \K\ form
associated
to $G$.
In a complex coordinate system
$(Z)$ around $p$
$$G_{\alpha\beta}=
2G(\frac{\partial}{\partial Z_{\alpha}},
\frac{\partial}{\partial Z_{\beta}})
=\frac{{\partial}^2\Phi}
{\partial Z_{\alpha}\partial\bar Z_{\beta}}.$$
 
A \K\ potential is not unique:
it is defined up to the sum with
the real part of a holomorphic function.
By duplicating the variables $Z$ and $\bar Z$
a potential $\Phi$ can be complex analytically
continued to a function 
$\tilde\Phi$ defined in a neighborhood
$U$ of the diagonal containing
$(p, \bar p)\in N\times\bar N$
(here $\bar N$ denotes the manifold
conjugated of $N$).
The {\em diastasis function} is the 
\K\ potential $D_p$
around $p$ defined by
$$D_p(q)=\tilde\Phi (q, \bar q)+
\tilde\Phi (p, \bar p)-\tilde\Phi (p, \bar q)-
\tilde\Phi (q, \bar p).$$
Among all the potentials the diastasis
is characterized by the fact that 
in every coordinates system 
$(Z)$ centered in $p$
$$D_p(Z, \bar Z)=\sum _{|j|, |k|\geq 0}
a_{jk}Z^j\bar Z^k,$$
with 
$a_{j 0}=a_{0 j}=0$
for all multi-indices
$j$.
The following proposition shows
the importance of
the diastasis
in the context of holomorphic
maps between 
\K\ manifolds.
\begin{prop}{\bf (Calabi)}\label{calabidiastasis}
Let 
$\varphi :(M, g)\rightarrow
(N, G)$
be a holomorphic and isometric
embedding between \K\ manifolds
and suppose that $G$ is real analytic.
Then $g$ is real analytic
and for every point 
$p\in M$
$$\varphi (D_p)=D_{\varphi (p)},$$
where $D_p$ (resp. $D_{\varphi (p)}$)
is the distasis of $g$ relative to
$p$ (resp. of $G$ relative to $\varphi (p)$). 
\end{prop}

We are now in the position
to state the second condition 
\newline
{\em Condition $A$: there exists
a point $p_*\in X=N\setminus Y$ such that
the diastasis $D_{{p_*}}$ is globally
defined and non-negative on $X$.} 

In order to describe the third condition
we need to introduce the concept of
Bochner's coordinates (cfr. \cite{boc}, \cite{ca}, \cite{hu},
\cite{hu1}, \cite{ti}, \cite{ru}).
Given a real analytic
\K\ metric $G$ on $N$
and a point $p\in N$,
one can always find local 
(complex) coordinates
in a neighborhood of 
$p$
such that
$$D_p(Z, \bar Z)=|Z|^2+
\sum _{|j|, |k|\geq 2}
b_{jk}Z^j\bar Z^k,$$
where 
$D_p$ is the diastasis
relative to $p$.
These coordinates,
uniquely defined up
to a unitary transformation,
are called 
{\em the Bochner's coordinates}
with respect to the point $p$.

One important feature of this
coordinates which we are going to use
in the proof of our main theorem
is the following:
\begin{teor}({\bf Calabi})\label{calabibochner}
Let 
$\varphi :(M, g)\rightarrow
(N, G)$
be a holomorphic and isometric
embedding between \K\ manifolds
and suppose that $G$ is real analytic.
If $(z_1,\dots ,z_m)$
is a system of Bochner's coordinates in
a neighborhood $U$
of $p\in M$ then there exists
a system of 
Bochner's coordinates
$(Z_1,\dots ,Z_n)$
with respect to $\varphi (p)$
such that
\begin{equation}\label{zandZ}
Z_1|_{\varphi(U)}=z_1,\dots ,Z_m|_{\varphi (U)}=z_m.
\end{equation}
\end{teor}


\noindent
{\em Condition $B$: the Bochner's 
coordinates with respect to the point
$p_*\in X$, given by the previous condition $(A)$,
are globally defined on 
$X$.}


Once that the previous
conditions $1$, $(A)$ and $(B)$ are satisfied, 
the obstruction to the existence
of K-E submanifolds $M$ of non-positive scalar curvature comes 
from comparing the volume of $M$ with the ``euclidean'' volume
of the ``affine part'' $\varphi (M) \cap X$, which, as we will see, is forced to be 
infinite.

Our first result is then the following:
\begin{teor}\label{teormain}
Let 
$N$ be 
a smooth
projective 
compactification 
of $X$ 
such that
$X$
is algebraically
biholomorphic to
${\complex}^n$
and let $G$
be a real analytic \K\
metric on $N$
such that the following two 
conditions are satisfied:
\begin{itemize}
\item [(A)]
there exists
a point $p_*\in X$ such that
the diastasis $D_{{p_*}}$ is globally
defined and non-negative on $X$;
\item [(B)]
the Bochner's 
coordinates with respect to 
$p_*$
are globally defined on 
$X$.
\end{itemize}
Then any
K--E sub-manifold
$(M, g)$ 
of $(N, G)$ has positive
scalar curvature.
\end{teor}


It would be interesting 
to drop the assumption that $X$
is algebraically biholomorphic to ${\complex}^n$ 
so to apply the previous result to any compactification of
${\complex}^n$ .

The algebraicity of the equivalence between $X$ and 
${\complex}^n$ comes in our argument only to estimate the euclidean volume
of the affine part of $M$. This leads to the following problem,
which we believe to be very interesting on its own sake:
\begin{problem}
Let $(N,Y)$ be a compactification of ${\complex}^n$, $M$ an 
analytic subvariety of pure dimension $m$, and $\pi \colon M\setminus
(M\cap Y) \rightarrow {\complex}^m$ an open projection. Is 
$vol(\pi(M\setminus (M\cap Y)))$ infinite?
Or even stronger, is the complement of $\pi(M\setminus (M\cap Y))$ 
an analytic subvariety?
\end{problem}

When $\pi$ is algebraic and $N$ is projective, this is a well known 
result of Chevalley (see Theorem 13.2 in \cite{bo}).
On the other hand, it is worthwhile pointing out that 
all known examples (even the non-projective ones!) satisfy our algebraicity
assumption. In fact, it has been conjectured by Peternell-Schneider
(Conjecture $5$ in \cite{ps}) that this is always true.

About the smoothness assumption of 
$N$, we should remark that 
for $n\leq 3$
smooth implies projective
(see \cite{ps}). 
It is also important
to observe that if also
$Y$ were smooth
in $N$ then
$N$ would be the projective
space and $Y$ a linear
subspace for $n\leq 6$
(and conjecturally 
for any $n$, see 
\cite{ps}).

As an application to the
theory of compactification
of ${\complex}^n$, 
we point out the following:
\begin{teor}\label{corolmain}
Let $N$
be a smooth 
compactification of
$X$  endowed with a 
a real
analytic K--E metric
$G$.
Suppose that
the following two conditions
are satisfied: 
\begin{itemize}
\item [(A)]
there exists
a point $p_*\in X$ such that
the diastasis $D_{{p_*}}$ is globally
defined and non-negative on $X$;
\item [(B1)]
the Bochner's coordinates
$(Z_1,\dots ,Z_n)$
around $p_*$
extend to holomorphic
functions
$(f_1,\dots ,f_n)$
on $X$.
\end{itemize}
Then
$c_1(N)>0$.
\end{teor}
Observe that in the previous theorem
we do not suppose as in Theorem \ref{teormain}
that $N$ is projective and that 
$X$ is biholomorphic
to ${\complex}^n$
in a  
algebraic way.
Moreover condition $(B1)$
is weaker 
than condition $(B)$ 
in Theorem \ref{teormain}
since the 
holomorphic functions
$(f_1,\dots ,f_n)$ 
are not necessarily
coordinates on $X$.

One should compare 
the previous theorem
with the following theorem
of Kodaira
which goes
deeply to the heart
of compactifications
of ${\complex}^n$ (see \cite{kod} and
\cite{ps}):
\begin{teor}{\bf (Kodaira)}
Let $(N, Y)$
be a smooth projective
compactification of
${\complex}^n$
with irreducible
$Y$. Then $N$
is a Fano manifold,
i.e. $c_1(N)>0$.
\end{teor}

The argument of the proofs of Theorems 
\ref{teormain} and \ref{corolmain}
give in fact the following:

\begin{teor}\label{senzacomp}
Let $N$ be a complex manifold
equipped with a real analytic K--E
metric $G$. Let $p_*\in N$ and let $U$
be a neighborhood of $p_*$ endowed with 
Bochner's coordinates $(Z_1,\dots ,Z_n)$.
Suppose that the following  two conditions
are satisified:
\begin{itemize}
\item [(A1)]
the diastasis $D_{p_*}$
is non-negative on $U$;
\item [(B2)]
the Euclidean volume of $U$ with respect to
the Bochner's coordinates is infinite namely:
$$\int _U\frac{i^n}{2^n}
dZ_1\wedge d\bar Z_1\wedge\dots\wedge 
dZ_n\wedge d\bar Z_n=\infty .$$
Then $c_1(N)>0$.
\end{itemize}
\end{teor}

In Section \ref{condA}
we wil show how condition
$(A)$ is related to geometric 
properties of a class of metrics,
called polarized metrics, studied by Tian \cite{ti}.
On the other hand conditions $(B), (B1)$
and $(B2)$ seem more difficult to be treated. 
For example we will see
that either the flat torus 
either a Riemann surface of genus $g\geq 2$
endowed with the hyperbolic metric
satisfy all the conditions of Theorem \ref{senzacomp}
on a open dense subset
except condition $(B2)$ and they have non-positive
first Chern class.

\vspace{0.5cm}
{\bf Acknowledgments}
During the preparation of
this paper we benefitted
from many conversations with
L. Alessandrini and G. Bassanelli
for which we  
thank them.
We also wish to thank 
A. Sambusetti for many stimulating
discussions.



\section{Comments on the assumptions}\label{condA}
The easiest example of compactification of
${\complex}^n$ which satisfies condition $(A)$
is given by 
${\complex}P^n={\complex}^n\cup
Y$ enodowed with the Fubini--Study metric
$g_{FS}$,
where
$Y={\complex}P^{n-1}$
is the hyperplane 
$Z_0=0$.
Indeed the diastasis
with respect to
$p_*=[1, 0,\dots ,0]$
is given by:
$$D_{p_*}(u, \bar{u})=
\log (1+\sum _{j=1}^{n}|u_j|^2).$$
where $(u_1,\dots ,u_n)$ are the affine coordinates, 
namely 
$u_j=\frac{Z_j}{Z_0}, j=1,\dots n$.


Another example of compactification of
${\complex}^n$
satisfying condition
$(A)$ 
is given by
the product of two complex projective spaces
$N={\complex}P^r\times {\complex}P^s, r+s=n$
endowed with the sum
of the Fubini--Study metrics
on ${\complex}P^r$
and ${\complex}P^s$
respectively
(what follows obviously works for a
product of finitely many projective spaces).
In this case we take as 
$Y=({\complex}P^{r-1}\times {\complex}P^s)
\cup ({\complex}P^r\times {\complex}P^{s-1})$
where
${\complex}P^{r-1}=\{Z_0=0\}$
and 
${\complex}P^{s-1}=\{W_0=0\}$.
Here we are equipping 
$N$ with the homogeneous
coordinates
$[Z_0,\dots ,Z_r]\times
[W_0,\dots ,W_s]$.
The diastasis 
with respect to the point
$p_*=
[1,\dots ,0]\times
[1,\dots ,0]$
is given by
$$D_{p_*}(u, \bar u)=
\log (1+\sum _{l=1}^{r+s}|u_l|^2),$$
where 
$u_j=\frac{Z_j}{Z_0}, j=1,\dots r$
and
$u_{r+k}=\frac{W_k}{W_0}, k=1,\dots s$.

We also remark that, by Proposition \ref{calabidiastasis},
condition $(A)$ is satisfied also by all
the \K\ submanifolds of the previous examples.

We know generalize the previous three examples
to a broader class of \K\ metrics called 
{\em polarized}. To describe this
let $L$ be a holomorphic line bundle 
on a compact complex manifold $N$.
A \K\ metric on $N$ is
{\em polarized} with 
respect to $L$   if the \K\
form $\Omega $
associated to $G$ 
represents the Chern class $c_{1}(L)$
of $L$.
The line bundle
$L$ is called a {\em polarization}
of $(N, G)$.
In terms of  cohomology
classes, a \K\ manifold
$(N, G)$ admits a
polarization
if and only if
$\Omega$ is integral,
i.e. its cohomology class
$[\Omega ]_{dR}$ in the de Rham group, 
is in the image of the natural map
$H^2(M,{\intero})\hookrightarrow H^2(M,{\complex})$.
The integrality of $\Omega $ implies,
by a well-known theorem of Kodaira, that
$N$ is a projective algebraic manifold.
This mean that $N$ admits a holomorphic embedding
into some complex projective space
${\complex}P^N$.
In this case a
polarization $L$ of 
$(N, G)$ is given by the 
restriction to $N$ of the hyperplane line
bundle on   
${\complex}P^N$.
Given any polarized \K\ metric $G$ on
$N$ 
one can find a  
hermitian metric $h$
on $L$ with its Ricci curvature form
$\Ric (h)=\Omega$ (see Lemma $1.1$ in \cite{ti}).

For each positive 
integer $k$, we denote by $L^{\otimes k}$
the $k$-th tensor power of $L$.
It
is a polarization
of the \K\ metric
$kG$ and  the hermitian
metric $h$ 
induces a
natural hermitian $h^k$
metric on $L^{\otimes k}$
such that 
$\Ric (h^k)=k\Omega$.

Denote by
$H^0(N, L^{\otimes k})$
the space of global holomorphic
sections of $L^{\otimes k}$.
It is in a natural way 
a complex Hilbert space  
with respect to the norm
$$\| s\|_{h^k}=
\langle s, s\rangle _{h^k}=
\int_{N}^{}h^k(s(x), s(x))
\frac{\Omega ^{n}(x)}{n!}<\infty,$$
for $s\in H^0(N, L^{\otimes k})$.


For sufficiently
large $k$ we can define
a holomorphic embedding
of $N$ into 
a complex projective space as follows.
Let  $(s_{0}, \dots , s_{d_k})$, 
be a orthonormal basis
for 
$H^0(N, L^{\otimes k})$
and let  $\sigma :U\rightarrow L^{\otimes k}$
be a trivialising
holomorphic
section  on the open set $U\subset N$.
Define the map
\begin{equation}\label{locals}
\varphi_\sigma:U\rightarrow 
{\complex}^{d_k+1}\setminus \{0\}, x\mapsto
\left(\frac{s_{0}(x)}{\sigma (x)}, 
\dots, \frac{s_{d_k}(x)}{\sigma (x)}\right).
\end{equation}

If $\tau :V\rightarrow L$
is another holomorphic trivialisation then
there exists a non-vanishing holomorphic function
$f$ on $U\cap V$ such that $\sigma(x)=f(x)\tau(x)$.
Therefore
one can define a
holomorphic  map 
\begin{equation}\label{phik}
\varphi _{k}:
N\rightarrow {\complex}P^{d_k},
\end{equation}
whose local expression in 
the open set $U$ is given by (\ref{locals}). 
It follows by the above mentioned
Theorem of Kodaira that,
for $k$ sufficiently large,
the map $\varphi _k$
is an embedding into
${\complex}P^{d_k}$
(see, e.g. \cite{gh}
for a proof).

The Fubini--Study metric
on  
${\complex}P^{d_k}$,
which we denote by $g^k_{FS}$,
restricts to a \K\
metric 
$G_k=\varphi_k^*g^{k}_{FS}$
on $N$ which is 
polarized 
with respect to $L^{\otimes k}$.
In \cite{ti}
Tian christined the set of normalized metrics $\frac{1}{k}G_k$ as
the {\em Bergmann} metrics on $N$ with respect 
to $L$ and he proves that
the sequence 
$\frac{1}{k}G_k$ converges to the metric
$G$ in the $C^2$-topology (see Theorem $A$ in \cite{ti}).
This theorem was further generalizes in the 
$C^{\infty}$ case by Ruan \cite{ru} (see also \cite{lu} and \cite{ze}).

The mentioned results have found important
applications to the existence problem
of K--E metrics. We now show that they also
imply interesting conditions on the diastasis of any 
polarized \K\ metric.

\begin{prop}\label{propCD}
Let $N$
be a complex manifold
endowed with a real analytic
\K\ metric $G$ polarized 
with respect to the line bundle $L$.
Let $(s_0,\dots ,s_d)$
be a othonormal basis for 
$H^0(M, L)$ and let $D=\{s_0=0\}$ be a divisor 
associated to $L$.
Suppose that the following two conditions
hold:
\begin{itemize}
\item [(C)]
there exists a point $p_*\in N\setminus D$
such that $s_j(p_*)=0, \forall j=1,\dots , d$;
\item [(D)]
there exists a natural number 
$k_0$ such that for all $k\geq k_0$
$s_0^{\otimes k}$  
is $L^2$-orthogonal to $s_0^{\otimes j_0}\otimes\dots\otimes s_d^{\otimes j_d}$,
$j_0+\dots +j_d=k, j_0\neq k$ with respect to the $L^2$-product induced
by $h^k$.
\end{itemize}
Then $D_{p_*}$, the diastasis of $G$ with respect to $p_*$,
is globally defined on $N\setminus D$ and non-negative.
\end{prop}
\dimostr
Let us denote by
$D^k_{p^*}$ the diastasis
of the metric $G_k=\varphi _k^*g^k_{FS}$ with respect to 
the point $p_*$.
First of all observe that for $k\geq k_0$, the space $H^0(M, L^{\otimes k})$
is genetated by 
$s_0^{\otimes j_0}\otimes\dots\otimes s_d^{\otimes j_d}$,
$j_0+\dots +j_d=k$ (cf. e.g. \cite{gh}). Thus,
if conditions
$(C)$ and $(D)$ hold, one can choose the map $\varphi _k$
in such a way that $\varphi _k(p_*)=[1, 0,\dots 0]\in {\complex}P^{d_k}$.
This implies that
$D=\varphi _k^{-1}(Y_k), \forall k\geq k_0$
where $Y_k={\complex}P^{d_k-1}$
is the hyperplane $Z_0=0$ in ${\complex}P^{d_k}$.
Since the diastasis of $({\complex}P^{d_k}, g^k_{FS})$
with respect to $[1, 0,\dots ,0]$ is globally defined on
${\complex}P^{d_k}\setminus Y_k$,
it follows by Proposition \ref{calabidiastasis}
that $D^k_{p^*}$ is globally defined on 
$N\setminus D$.
By the very definition of diastasis
in any coordinate system
$(Z_1,\dots ,Z_n)$
centered in $p_*$
we have:
$D^k_{p^*}=\sum _{|i|,|j|\geq 0}
a_{ij}(k)Z^i\bar Z^j$
where $a_{ij}(k)$
is a matrix depending on 
$k$ such that
$a_{j0}(k)=a_{0j}(k)=0$.
This is equivalent to
\begin{equation}\label{dpotential}
\frac{\partial D^k_{p^*}}
{\partial Z^l}(p_*)=
\frac{\partial D^k_{p^*}}
{\partial\bar Z^m}(p_*)=0,
\end{equation}
for any  multi-indices
$l, m$ and for any
$k$.
By the above mentioned theorem of Ruan  
the sequence
$\frac{D^k_{p_*}}{k}$ $C^{\infty}$-converges
to a \K\ potential
$\Phi$ for the metric $G$
(around the point $p$)
which is then globally defined
on $N\setminus D$. Since $D^k_{p_*}$
is non-negative on $N\setminus D$, being
the restriction of the diastasis of the Fubini--Study metric
on ${\complex}P^{d_k}$, the potential $\Phi$
is non-negative on $N\setminus D$. 
Moreover property (\ref{dpotential})
goes to the limit, namely
$$\frac{\partial \Phi}
{\partial Z^l}(p_*)=
\frac{\partial \Phi}
{\partial\bar Z^m}(p_*)=0,$$
for any  multi-indices $l, m$.
Since $(Z_1,\dots ,Z_n)$
is an arbitrary coordinate system 
centered in $p_*$ it follows, again by the definition
of diastasis, that 
$\Phi =D_{p_*}$.
\fdim

\begin{remar}\normalshape
Observe that conditions $(C)$ and 
$(D)$ of Proposition \ref{propCD}
are satisfied by 
$({\complex}P^n, g_{FS})$.
Indeed  the polarization bundle $L$ of
${\complex}P^n$ is given by the
hyperplane bundle and the space
of holomorphic sections of $L^{\otimes k}$
can be identified by
the homogeneous polynomials of degree $k$ in 
the variables
$(Z_0,\dots , Z_N)$ (see e.g. \cite{gh}).
A simple calculation shows that
$Z_0^{j_0}\cdots Z_N^{j_N}, 
j_0+\cdots +j_N=k$ is indeed an orthonormal basis
for $H^0({\complex}P^n, L^{\otimes k})$ and hence
conditions $(C)$ and $(D)$ are 
satisfied with respect 
to the point $p_*=[1, 0,\dots , 0]$.
We also observe that conditions $(C)$ and $(D)$
hold
for the product of two complex projective
spaces (or a finite number of projective spaces)
$N={\complex}P^r\times {\complex}P^s, r+s=n$ endowed,
as in the beginning of this section, with the sum of
the Fubini--Study metrics on the two factors.
In this case the polarization bundle $L$ on $N$ 
is given by the pull-back of the hyperplane
bundle on ${\complex}P^{rs+r+s}$ via the Segree embedding.
Moreover, a straightforward calculation
shows that a orthonormal basis for the space $H^0(N, L^{\otimes k})$
is given by   
$Z_0^{j_0}\cdots Z_r^{j_r}W_0^{l_0}\cdots W_s^{l_s}, 
j_0+\cdots +j_r=k, l_0+\cdots +l_s=k$,
and hence conditions $(C)$ and
$(D)$ hold true with respect to the point
$p_*=[1, 0,\dots ,0]\times[1, 0,\dots , 0]$.
\end{remar}

We now discuss conditions $(B), (B1)$ and $(B2)$. 
Condition $(B)$ is clearly satisfied by the
above examples. Indeed in these cases the Bochner's coordinates
with respect to the point $p_*$
are given by the affine coordinates.

By Theorem \ref{calabibochner},
condition $(B1)$
it is satisfied 
by any \K\ submanifold of the complex projective
space.

We finally want to point out that condition $(B2)$
in Theorem \ref{senzacomp} is necessary.
Let $N={\complex}^n/{\intero}^{2n}$ be the
$n$-dimensional complex torus
with the flat metric 
$G=\frac{i}{2}\sum _{j=1}^{n}|dz_j|^2$ coming from
${\complex}^n$. This metric is polarized
by the so called theta bundle.
Let $p_*$ be the origin in ${\complex}^n$
and let $U$  be
a fundamental domain containing $p_*$
The diastasis $D_{p_*}$ of $G$ with respect to
$p_*$ reads as 
$D_{p_*}(z, \bar z)=\sum _{j=1}^{n}|z_j|^2$
and the euclidean coordinates $(z_1,\dots ,z_n)$
are in fact Bochner's coordinates around $p_*$.
Therefore $U$ is
an open dense subset  of $N$
for which all the properties of Theorem \ref{senzacomp}
are satisfied but, of course, $U$ has finite 
euclidean volume.
The same argument can be applied to any Riemann
surface of genus $g\geq 2$ endowed with the
hyperbolic metric which is polarized by the 
canonical bundle.
Therefore these examples, having non-positive
first Chern class show that the assumption
$(B2)$ is necessary.



\section{Proof of the main results}
{\bf Proof of Theorem \ref{teormain}:}


Let 
$p$
be a point
in $M$ 
such that 
$\varphi (p)=p_*$,
where $p_*$ is the point in 
$N$ given by condition $(A)$.
Take Bochner's 
coordinates 
$(z_1,\dots ,z_m)$
in a neighborhood 
$U$ of 
$p$ which we 
take small enough to
be contractible.
Since the \K\ metric
$g$ is Einstein with
(constant) scalar
curvature $s$ then:
$\rho_{\omega}=\lambda \omega$
where 
$\lambda$ is the Einstein
constant, i.e. 
$\lambda=\frac{s}{2m}$,
and $\rho_{\omega}$
is the \emph{Ricci form}.
If $\omega =\frac{i}{2}
\sum _{j=1}^{m}g_{j\bar{k}}
dz_{j}\wedge d\bar{z}_{\bar{k}}$
then 
$\label{rholocal}
\rho _{\omega}=-i\partial
\bar{\partial}\log \det g_{j\bar{k}}$
is the local expression of
its \emph{Ricci form}.

Thus the volume form
of $(M, g)$ reads
on 
$U$ as:
\begin{equation}\label{voleucl}
\frac{\omega^m}{m!}=\frac{i^m}{2^m}
e^{-\frac{\lambda}{2}D_p+F+\bar F}
dz_1\wedge d\bar z_1\wedge\dots\wedge 
dz_m\wedge d\bar z_m\, \, ,
\end{equation}
where $F$ is a holomorphic
function on $U$
and
$D_p=\varphi^{-1}(D_{p_*})$
is the diastasis on $p$
(cfr. Proposition
\ref{calabidiastasis}).
 
We claim that
$F+\bar F=0$.
Indeed, 
observe that 
$$\frac{\omega ^m}{m!}=\frac{i^m}{2^m}
\det (\frac{\partial ^2 D_p}
{\partial z_{\alpha} \partial \bar z_{\beta}})
dz_1\wedge d\bar z_1\wedge\dots\wedge 
dz_m\wedge d\bar z_m .$$
By the very definition
of Bochner's coordinates it is easy
to check that 
the expansion of 
$\log\det (\frac{\partial ^2 D_p}
{\partial z_{\alpha} \partial \bar z_{\beta}})$
in the 
$(z,\bar z)$-coordinates
contains only mixed terms
(i.e. of the form
$z^j\bar z^k, j\neq 0, k\neq 0$).
On the other hand
by formula  (\ref{voleucl})
$$-\frac{\lambda}{2} D_p + F + \bar F=
\log\det (\frac{\partial ^2 D_p}
{\partial z_{\alpha} \partial \bar z_{\beta}}).$$

Again by the defintion of the Bochner's 
coordinates this forces
$F + \bar F$ to be zero,
proving our claim.

By Theorem \ref{calabibochner}
there exist Bochner's
coordinates 
$(Z_1, \dots , Z_n)$
in a neighborhood of 
$p_*$
satisfying 
(\ref{zandZ}).
Moreover, by condition $(B)$
this coordinates 
are 
globally defined
on
$X$.
Hence, by 
formula (\ref{voleucl})
(with $F+\bar F=0$),
the
$m$-forms 
$\frac{\Omega^m}{m!}$
and
$e^{-\frac{\lambda}{2}D_{p_*}}
dZ_1\wedge d\bar Z_1\wedge\dots\wedge 
dZ_m\wedge d\bar Z_m$ 
globally defined on $X$
agree on the open set
$\varphi (U)$.
Since they are
real analytic  they
must agree
on the
connected open set
$\hat M=\varphi (M)\cap X$, i.e.
\begin{equation}\label{eqforms}
\frac{\Omega^m}{m!}=\frac{i^m}{2^m}
e^{-\frac{\lambda}{2}
D_{p_*}}
dZ_1\wedge d\bar Z_1\wedge\dots\wedge 
dZ_m\wedge d\bar Z_m.
\end{equation}

Since 
$\frac{\Omega^m}{m!}$
is a volume form on $\hat M$
we deduce that
the restriction of the
projection map
$$\pi :X\cong{\complex}^n
\rightarrow {\complex}^m:
(Z_1,\dots ,Z_n)\mapsto 
(Z_1,\dots ,Z_m)$$
to $\hat M$ 
is open.
Since it
is also algebraic 
(because the biholomorphism
between $X$ and  ${\complex}^n$
is algebraic),
its image 
contains a Zariski open
subset of ${\complex}^m$
(see Theorem
$13.2$ in \cite{bo}),
hence 
its euclidean volume,
$\vol_{eucl}(\pi (\hat M))$,  
has to be infinite.

Suppose now
that the scalar curvature
of $g$ is non-positive.
By formula 
(\ref{eqforms})
and  by the fact that $D_{p_*}$ is non-negative,
we get 
$\vol (\hat M, g)\geq
\vol_{eucl}(\pi (\hat M))$
which is the desired contradiction,
being the volume of $M$
(and hence that of 
$\hat M$)
finite.
\fdim


{\bf Proof of Theorem \ref{corolmain}:} 
Since the metric $G$
is K--E, working 
as in Theorem
\ref{teormain} one gets that 
\begin{equation}\label{voleucl2}
\frac{\Omega^n}{n!}=\frac{i^n}{2^n}
e^{-\frac{\lambda}{2}D_{p_*}}
dZ_1\wedge d\bar Z_1\wedge\dots\wedge 
dZ_n\wedge d\bar Z_n\, \, ,
\end{equation}
on a contractible open set $U$ where
the Bochner's coordinates
$(Z_1,\dots ,Z_n)$ are defined.
Let $(f_1,\dots , f_n)$
be holomorphic functions
on $X$ which extend 
$(Z_1,\dots ,Z_n)$
(given by condition $(B1)$).
Then
the
$n$-forms 
$\frac{\Omega^n}{n!}$
and
$e^{-\frac{\lambda}{2}D_{p_*}}
df_1\wedge d\bar f_1\wedge\dots\wedge 
df_n\wedge d\bar f_n$ 
globally defined on $X$
agree on the open set $U$.
Since they are
real analytic  they
must agree
on 
$X$, i.e.
\begin{equation}\label{eqforms2}
\frac{\Omega^n}{n!}=\frac{i^n}{2^n}
e^{-\frac{\lambda}{2}
D_{p_*}}
df_1\wedge d\bar f_1\wedge\dots\wedge 
df_n\wedge d\bar f_n.
\end{equation}

As a consequence
the map
$$\pi :X\cong{\complex}^n
\rightarrow {\complex}^n:
x\mapsto 
(f_1(x),\dots ,f_n(x))$$
is open.
Moreover, since it is
holomorphic it is also surjective
and hence
$\vol_{eucl}(\pi (X))$  
is infinite.

Suppose
that the scalar curvature
of $G$ is non-positive.
By formula 
(\ref{eqforms2})
and by the fact that
$D_{p_*}$
is non-negative,
one gets that 
$\vol (X)\geq
\vol_{eucl}(\pi (X))$
which is the desired contradiction,
being the volume of $X$
finite.
\fdim





\begin{thebibliography}{99}

\bibitem{boc} S. Bochner,
{\em Curvature in Hermitian metric},
Bull. Amer. Math. Soc. 53 (1947), 179-195. 


\bibitem{bo} A. Borel,
{\em Linear algebraic groups},
second ed., GTM n. 126
Springer--Verlag, New-York (1991). 


\bibitem{ca} E. Calabi,
{\em Isometric Imbeddings of Complex Manifolds},
Ann. of Math. 58 (1953), 1-23. 

\bibitem{cgr1} M. Cahen, S. Gutt, J. H. Rawnsley,
{\em Quantization of \K\ manifolds I:
Geometric interpretation of Berezin's quantization},
J. Geom. Phys. 7 (1990), 45-62.


\bibitem{ch} S. S. Chern,
{\em On Einstein hypersurfaces in a \K\
manifold of constant sectional curvature},
J. Differ. Geom. 1 (1967), 21-31.

\bibitem{gh} P. Griffiths and J. Harris,
{\em Principles of Algebraic Geometry},
John Wiley and Sons Inc. 1978.

\bibitem{hi} F. Hirzebruck,
{\em Some problems on differentiable
and complex manifolds},
Ann. of Math. 60 (1962), 213-236 . 

\bibitem{hu} D. Hulin,
{\em Sous-varietes complexes d'Einstein de
l'espace projectif},
Bull. Soc. math. France 124 (1996), 277-298.

\bibitem{hu1} D. Hulin,
{\em \K\--Einstein metrics and
projective embeddings},
J. Geom. Anal. 10 (2000), 525-528.

\bibitem{kod} K. Kodaira,
{\em Holomorphic mappings of polydiscs
into compact complex manifolds},
J. Differ. Geom. 6 (1971), 33-46.

\bibitem{lu} Z. Lu,
{\em On the lower terms of the asymptotic expansion of Tian--Yau--Zelditch},
Amer. J. Math. 122 (2000),  235-273.


\bibitem{ps} T. Peternell and M. Schneider,
{\em Compactification of ${\complex}^n $:
a survey},
in Several complex variables and complex geometry,
Proc. Symp. Pure Math vol. 52 part 2 (1991), 455-466.


\bibitem{ru} W. D. Ruan, 
{\em Canonical coordinates and Bergmann metrics},
Comm. in Anal. and Geom. (1998), 589-631.

\bibitem{Ta} M. Takeuchi,
{\em Homogeneous \K\ Manifolds in Complex
Projective Space},
Japan J. Math. vol. 4 (1978), 171-219.


\bibitem{ti} G. Tian, 
{\em On a set of polarized \K\
metrics on algebraic manifolds},
J. Differ. Geom. 32 (1990), 99-130.

\bibitem{ts} K. Tsukada,
{\em Einstein-\K\ Submanifolds with codimension
two in a Complex Space Form},
Math. Ann. 274 (1986), 503-516.

\bibitem{um} M. Umehara,
{\em Einstein-\K\ Submanifolds of a Complex Linear 
or Hyperbolic Space},
Tohoku  Math. J. 39 (1987), 385-389.

\bibitem{ze} S. Zeldtich, 
{\em Szeg\"{o} Kernel and a Theorem of Tian},
J. Diff. Geometry 32 (1990), 99-130.


\end{thebibliography}

 
 
 
 
\end{document}


math0103119.extracted_bib
\begin{thebibliography}{99}

\bibitem{boc} S. Bochner,
{\em Curvature in Hermitian metric},
Bull. Amer. Math. Soc. 53 (1947), 179-195. 


\bibitem{bo} A. Borel,
{\em Linear algebraic groups},
second ed., GTM n. 126
Springer--Verlag, New-York (1991). 


\bibitem{ca} E. Calabi,
{\em Isometric Imbeddings of Complex Manifolds},
Ann. of Math. 58 (1953), 1-23. 

\bibitem{cgr1} M. Cahen, S. Gutt, J. H. Rawnsley,
{\em Quantization of \K\ manifolds I:
Geometric interpretation of Berezin's quantization},
J. Geom. Phys. 7 (1990), 45-62.


\bibitem{ch} S. S. Chern,
{\em On Einstein hypersurfaces in a \K\
manifold of constant sectional curvature},
J. Differ. Geom. 1 (1967), 21-31.

\bibitem{gh} P. Griffiths and J. Harris,
{\em Principles of Algebraic Geometry},
John Wiley and Sons Inc. 1978.

\bibitem{hi} F. Hirzebruck,
{\em Some problems on differentiable
and complex manifolds},
Ann. of Math. 60 (1962), 213-236 . 

\bibitem{hu} D. Hulin,
{\em Sous-varietes complexes d'Einstein de
l'espace projectif},
Bull. Soc. math. France 124 (1996), 277-298.

\bibitem{hu1} D. Hulin,
{\em \K\--Einstein metrics and
projective embeddings},
J. Geom. Anal. 10 (2000), 525-528.

\bibitem{kod} K. Kodaira,
{\em Holomorphic mappings of polydiscs
into compact complex manifolds},
J. Differ. Geom. 6 (1971), 33-46.

\bibitem{lu} Z. Lu,
{\em On the lower terms of the asymptotic expansion of Tian--Yau--Zelditch},
Amer. J. Math. 122 (2000),  235-273.


\bibitem{ps} T. Peternell and M. Schneider,
{\em Compactification of ${\complex}^n $:
a survey},
in Several complex variables and complex geometry,
Proc. Symp. Pure Math vol. 52 part 2 (1991), 455-466.


\bibitem{ru} W. D. Ruan, 
{\em Canonical coordinates and Bergmann metrics},
Comm. in Anal. and Geom. (1998), 589-631.

\bibitem{Ta} M. Takeuchi,
{\em Homogeneous \K\ Manifolds in Complex
Projective Space},
Japan J. Math. vol. 4 (1978), 171-219.


\bibitem{ti} G. Tian, 
{\em On a set of polarized \K\
metrics on algebraic manifolds},
J. Differ. Geom. 32 (1990), 99-130.

\bibitem{ts} K. Tsukada,
{\em Einstein-\K\ Submanifolds with codimension
two in a Complex Space Form},
Math. Ann. 274 (1986), 503-516.

\bibitem{um} M. Umehara,
{\em Einstein-\K\ Submanifolds of a Complex Linear 
or Hyperbolic Space},
Tohoku  Math. J. 39 (1987), 385-389.

\bibitem{ze} S. Zeldtich, 
{\em Szeg\"{o} Kernel and a Theorem of Tian},
J. Diff. Geometry 32 (1990), 99-130.


\end{thebibliography}