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abstract: |
    [Let $Y$ be a regular covering of a complex projective manifold $M\hookrightarrow\Co\P^{N}$ of dimension $n\geq 2$. Let $C$ be intersection with $M$ of at most $n-1$ generic hypersurfaces of degree $d$ in $\Co\P^{N}$. The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $H^{\infty}(Y)$ and $H^{\infty}(X)$ be the Banach spaces of bounded holomorphic functions on $Y$ and $X$ in the corresponding supremum norms. We prove that the restriction $H^{\infty}(Y)\longrightarrow H^{\infty}(X)$ is an isometry for $d$ large enough. This answers the question posed in \[L\].\
    ]{}
author:
- |
    Alexander Brudnyi[^1]\
    Department of Mathematics and Statistics\
    University of Calgary\
    Calgary, Canada
title: 'A Uniqueness Property for $H^{\infty}$ on Coverings of Projective Manifolds'
---

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=msbm10 scaled 1 =msbm7 scaled 1 =msbm5 scaled 1 === ¶[[P]{}]{} \[section\] \[Th\][Lemma]{} \[Th\][Corollary]{} \[Th\][Definition]{} \[Th\][Proposition]{} \[Th\][Remark]{} \[Th\][Example]{}

. Introduction and Formulation of the Result.
=============================================

[**1.1. An Extension Theorem.**]{}

Let $M$ be a complex projective manifold of -10000 dimension $n\geq 2$ with a Kähler form $\omega$ and let $L$ be a positive line bundle on $M$ with canonical connection $\nabla$ and curvature $\Theta$ in a hermitian metric $h$. Let $C$ be the common zero locus of holomorphic sections $s_{1},...,s_{k}$, $k<n$, of $L$ over $M$ which, in a trivialization, can be completed to a set of local coordinates at each point $C$. Then $C$ is a (possibly disconnected) $k$-dimensional submanifold of $M$ which will be referred to as an [*$L$-submanifold of $M$*]{}. Let $\pi:Y_{G}\longrightarrow M$ be a regular covering of $M$ with a transformation group $G$ and $X_{G}=\pi^{-1}(C)$. We denote the pullbacks to $Y_{G}$ of $\omega$ and $\Theta$ by the same letters.

\[e0\] [If $L$ is very ample, then it is pullback of the hyperplane bundle by an embedding of $M$ into some projective space $\Co\P^{N}$. Further, zero loci of holomorphic sections of $L$ are hyperplane sections of $M$. By Bertini’s theorem, the generic linear subspace of codimension $n-k$, $k<n$, intersects $M$ transversely in a smooth manifold $C$ of dimension $k$, and by the Lefschetz hyperplane theorem, $C$ is connected and the induced homomorphism $\pi_{1}(C)\longrightarrow \pi_{1}(M)$ of fundamental groups is surjective. Thus in this case $X_{G}\subset Y_{G}$ is a connected submanifold.]{}

Further, let $dist(\cdot,\cdot)$ be the distance on $Y_{G}$ induced by $\omega$. Consider a function $\phi: Y_{G}\longrightarrow\Re$ such that $d\phi$ is bounded, i.e. $$|\phi(x)-\phi(y)|\leq a\cdot dist(x,y)\  \ \ \ 
{\rm for\ some}\ \ \ a>0.$$ By ${\cal O}_{\phi}(X_{G})$ we denote the vector space of holomorphic functions on $X_{G}$ such that $|f|^{2}e^{-\phi}$ is integrable on $X_{G}$ with respect to the volume form of the induced Kähler metric on $X_{G}$. This is a Hilbert space with respect to the inner product $$(f,g)\mapsto\int_{X_{G}}f\overline{g}e^{-\phi}\omega^{k}\ .$$ We define ${\cal O}_{\phi}(Y_{G})$ similarly. By $|\cdot|_{\phi,X_{G}}$ and $|\cdot|_{\phi,Y_{G}}$ we denote the corresponding norms. It was shown in \[L\] that the restriction determines a continuous linear map $$\rho: {\cal O}_{\phi}(Y_{G})\longrightarrow {\cal O}_{\phi}(X_{G}),\ \ \
f\mapsto f|_{X_{G}}\ .$$ The following remarkable result was proved by Lárusson \[L, Th.1.2\].

\[la\] Suppose $$\Theta\geq i\partial\overline{\partial}\phi+\epsilon\omega$$ for some $\epsilon>0$ in the sense of Nakano. Then $\rho$ is an isomorphism.

[**1.2. Formulation of the Main Result.**]{} An important example of a function $\phi$ as above is obtained by smoothing the distance $\delta$ from a fixed point $o$ in $Y_{G}$. By a result of Napier \[N\], there is a smooth function $\tau$ on $Y_{G}$ such that

[(A)]{}

:   $c_{1}\delta\leq\tau\leq c_{2}\delta+c_{3}$ for some $c_{1},c_{2},c_{3}>0$,

[(B)]{}

:   $d\tau$ is bounded, and

[(C)]{}

:   $i\partial\overline\partial\tau$ is bounded.

Furthermore, by (A) and since the curvature of $Y_{G}$ is bounded below, there is $c>0$ such that $e^{-c\tau}$ is integrable on $Y_{G}$. Then $e^{-c\tau}$ is also integrable on $X_{G}$. We set $$\label{A}
A:=\frac{cc_{2}}{c_{1}}\ .$$

Let $\tilde L$ be any positive line bundle on $M$ with curvature $\tilde\Theta$. By (C) there is a non-negative integer $m_{0}$ such that for any integer $m>m_{0}$ $$\label{e1}
m\tilde\Theta> i\partial\overline\partial(A\tau)\ .$$ We set $L:=\tilde L^{\otimes m}$. Then Lárusson’s theorem holds for coverings $X_{G}:=\pi^{-1}(C)\subset Y_{G}$ of $L$-submanifolds $C\subset M$ with $\phi:=A\tau$ and with $\phi:=c\tau$ (because $A\geq c$).

Let $H^{\infty}(Y_{G}),\ H^{\infty}(X_{G})$ be the Banach spaces of bounded holomorphic functions on $Y_{G}$ and $X_{G}$ in the corresponding supremum norms.

\[te1\] The map $\rho: H^{\infty}(Y_{G})\longrightarrow H^{\infty}(X_{G})$, $f\mapsto f|_{X_{G}}$, is an isometry.

[**1.3. Corollaries and Examples.**]{} Let $X$ be a complex manifold and $H^{\infty}(X)$ be the Banach algebra (in the supremum norm) of bounded holomorphic functions on $X$. The maximal ideal space ${\cal M}={\cal M}(H^{\infty}(X))$ is the set of all nontrivial linear multiplicative functionals on $H^{\infty}(X)$. The norm of any $\phi\in {\cal M}$ is $\leq 1$ and so ${\cal M}$ is embedded into the unit ball of the dual space $(H^{\infty}(X))^{*}$. Then ${\cal M}$ is a compact Hausdorff space in the weak $*$ topology induced by $(H^{\infty}(X))^{*}$ (i.e. the [*Gelfand topology*]{}). Further, there is a continuous map $i:X\longrightarrow {\cal M}$ taking $x\in X$ to the evaluation homomorphism $f\mapsto f(x)$. This map is an emebedding if $H^{\infty}(X)$ separates points of $X$. Recall also that the complement to the closure of $i(X)$ in ${\cal M}$ is called [*the corona*]{}. The [*corona problem*]{} is to determine those $X$ for which the corona is empty. For example, according to Carleson’s celebrated Corona Theorem \[C\] this is true for $X$ being the open unit disk $\Di\subset\Co$. Also there are non-planar Riemann surfaces for which the corona is non-trivial (see e.g. \[JM\], \[G\], \[BD\] and references therein). The general problem for planar domains is still open, as is the problem in several variables for the ball and polydisk. In \[L, Th. 2.1\] Lárusson discovered simplest examples of Riemann surfaces with big corona. Namely, using his Theorem \[la\] he proved that if $Y_{G}\subset\Co^{n}$ is a bounded domain and $X_{G}\subset Y_{G}$ is a Riemann surface satisfying assumptions of Theorem \[te1\] then the natural map $i:X_{G}\hookrightarrow {\cal M}(H^{\infty}(X_{G}))$ extends to an embedding $Y_{G}\hookrightarrow  {\cal M}(H^{\infty}(X_{G}))$. The next statement is an extension of his result.

\[cor1\] Under the assumptions of Theorem \[te1\], the transpose map -10000 $\rho^{*}:{\cal M}(H^{\infty}(X_{G}))\longrightarrow 
{\cal M}(H^{\infty}(Y_{G}))$, $\phi\mapsto\phi\circ\rho$, is a homeomorphism.

This follows from the fact that $\rho: H^{\infty}(Y_{G})\longrightarrow
H^{\infty}(X_{G})$ is an isometry of Banach algebras.     $\Box$

\[e1\] [(1) (The references for this example are in \[L, Sect. 4\].) Let $M$ be a projective manifold covered by the unit ball $\B\subset\Co^{n}$ with a positive line bundle $L$ with curvature $\Theta$, and $X\subset\B$ be the preimage of an $L$-submanifold $C\subset M$. Let $\delta$ be the distance from the origin in the Bergman metric of $\B$. By $\omega$ we denote the Kähler form of the Bergman metric. It was shown in \[L, Sect. 4\] that there is a nonnegative function $\tau$ on $\B$ such that $i\partial\overline{\partial}\tau=\omega$, $d\tau$ is bounded, and $$\sqrt{n+1}\delta\leq\tau\leq\sqrt{n+1}\delta+(n+1)\log\ 2 .$$ Moreover, $$\int_{\B}e^{-c\tau}\omega^{n}<\infty\ \ \ \ {\rm if\ and\ only\ if}\ \ \
c>\frac{2n}{n+1}\ .$$ Applying Theorem \[te1\] (with $c_{2}=c_{1}=\sqrt{n+1}$) we obtain that [*$\rho:H^{\infty}(\B)\rightarrow H^{\infty}(X)$ is an isometry if $\Theta>\frac{2n}{n+1}\omega$.*]{} This holds for instance if $L=K^{\otimes m}$ with $m\geq 2$ where $K$ is the canonical bundle of $M$.\
(2) Let $S$ be a compact complex curve of genus $g\geq 2$ and $\Co\To$ be a one-dimensional complex torus. Consider an $L$-curve $C$ in $M:=S\times\Co\To$ with a very ample $L$ satisfying the assumptions of Theorem \[te1\]. Let $\pi:\Di\times\Co\longrightarrow M$ be the universal covering. Then Theorem \[te1\] is valid for the connected curve $X:=\pi^{-1}(C)\subset\Di\times\Co$. This implies that any $f\in H^{\infty}(X)$ is constant on each $S_{y}:=(\{y\}\times\Co)\cap X$, $y\in\Di$. Note that $S_{y}$ is union of the orbits of some $z_{iy}\in X$, $i=1,...,k$, under the natural action of the group $\pi_{1}(\Co\To)\ (\cong\Z\oplus\Z)$ on $\Di\times\Co$.]{}

. Proof of Theorem 1.3.
=======================

Let us fix a fundamental compact $K$ of the action of $G$ on $Y_{G}$, i.e., $Y_{G}=\cup_{g\in G}\ g(K)$. Consider finite covers ${\cal U}=(U_{i})$ and ${\cal V}=(V_{j})$ of $K$ by compact coordinate polydisks such that each $V_{j}$ belongs to the interior of some $U_{i_{j}}$.

\[le1\] Let $f$ be a holomorphic function defined in an open neighbourhood $O$ of $\cup_{i}\ U_{i}$. Assume that $$\int_{O}|f|^{2}\omega^{n}=B<\infty\ .$$ Then there is a constant $b>0$ (depending on ${\cal U}$ and ${\cal V}$ only) such that $$\label{e2}
\max_{K}|f|\leq b\sqrt{B}\ .$$

The proof of the lemma is the consequence of the following facts:\
(a) after the identification of $U_{i_{j}}$ with the closed unit polydisk $D$ and of $V_{j}$ with a compact subset $D_{j}\subset D$, the volume form $\omega^{n}$ restricted to each $U_{i_{j}}$ is equivalent to the Euclidean volume form $do:=dz_{1}\wedge d\overline{z}_{1}\wedge\dots\wedge
dz_{n}\wedge d\overline{z}_{n}$ ;\
(b) the Bergman inequality (see \[GR, Ch.6, Th.1.3\]) $$\max_{D_{j}}|f|\leq\frac{(\sqrt{n})^{n}}{(\sqrt{\pi}d)^{n}}\cdot
\left(\int_{D}|f|^{2}do\right)^{1/2},$$ where $d$ is the Euclidean distance from $D_{j}$ to the boundary of $D$.\
We leave the details to the reader.      $\Box$

Let $dist(\cdot,\cdot)$ be the distance on $Y_{G}$ in the metric induced by $\omega$. In particular, $\delta(x):=dist(x,o)$. Since $\omega$ is invariant with respect to the action of $G$ we also have $dist(g(x),g(y))=dist(x,y)$ for any $g\in G$. From inequalities (A) for $\tau$ and the triangle inequality for the distance we obtain $$\label{e3}
\begin{array}{c}
\tau(g(x))\geq c_{1}dist(g(x),o)\geq c_{1}[dist(g(x),g(o))-dist(g(o),o)]=\\
\\
c_{1}[dist(x,o)-dist(g(o),o)]\geq
(c_{1}/c_{2})\tau(x)-(c_{1}c_{3}/c_{2})-
c_{1}\delta(g(o))\ .
\end{array}$$ Further, if $x\in K$ then $$\label{e4}
a_{1}\leq\tau(x)\leq a_{2}\ \  {\rm for\ some}\ \ a_{1},a_{2}>0\ .$$

Below by $|\cdot|_{\infty, X_{G}}$ and $|\cdot|_{\infty, Y_{G}}$ we denote the corresponding $H^{\infty}$-norms. Let $f\in H^{\infty}(X_{G})$. Then $f\in {\cal O}_{A\tau}(X_{G})\cap {\cal O}_{c\tau}(X_{G})$ and there is $a_{3}>0$ such that $$\max\{|f|_{A\tau, X_{G}}, |f|_{c\tau, X_{G}}\}\leq  a_{3}\sup_{X_{G}}|f|:=
a_{3}|f|_{\infty, X_{G}}\ .$$ By Theorem \[la\], there is a unique $\tilde f\in {\cal O}_{A\tau}(Y_{G})\cap {\cal O}_{c\tau}(Y_{G})$ such that $\tilde f|_{X_{G}}=f$ and $$\max\{|\tilde f|_{A\tau, Y_{G}}, |\tilde f|_{c\tau, Y_{G}}\}\leq a_{4}
\max\{|f|_{A\tau, X_{G}}, |f|_{c\tau, X_{G}}\}\ \ {\rm for\ some}\
\ a_{4}>0 .$$ Combining these inequalities with (\[e4\]) and (\[e2\]) yields $$\max_{K}|\tilde f|\leq a_{5}|f|_{\infty, X_{G}},$$ with some $a_{5}>0$ depending on $X_{G},Y_{G}$ only. For a fixed $g\in G$ consider $(g^{*}f)(z):=f(g(z))$. As above, there is a unique function $\tilde f_{g}\in {\cal O}_{A\tau}(Y_{G})\cap {\cal O}_{c\tau}(Y_{G})$, $\tilde f_{g}|_{X_{G}}=g^{*}f$, such that $$\max_{K}|\tilde f_{g}|\leq a_{5}|f|_{\infty, X_{G}},$$ But according to (\[e3\]) and (\[A\]), function $(g^{*}\tilde f)(z):=\tilde f(g(z))$ belongs to ${\cal O}_{A\tau}(Y_{G})$ and $(g^{*}\tilde f-\tilde f_{g})|_{X_{G}}\equiv 0$. Thus by Theorem \[la\] we have $\tilde f_{g}=g^{*}\tilde f$. Since $K$ is the fundamental compact, the above inequality for each $\tilde f_{g}$ implies that $$\label{e5}
|\tilde f|_{\infty, Y_{G}}\leq a_{5}|f|_{\infty, X_{G}}\ .$$ We will prove now that $a_{5}=1$ which gives the required statement. Indeed, the same arguments as above show that for any integer $n\geq1$ the function $(\tilde f)^{n}$ is the unique extension of $f^{n}$ satisfying (\[e5\]): $$|(\tilde f)^{n}|_{\infty, Y_{G}}\leq a_{5}|f^{n}|_{\infty, X_{G}}\ .$$ Thus $$|\tilde f|_{\infty, Y_{G}}\leq 
\lim_{n\to\infty}(a_{5})^{1/n}|f|_{\infty, X_{G}}=|f|_{\infty, X_{G}}$$ The proof of the theorem is complete.      $\Box$

[ ]{} D. E. Barett and J. Diller, A new construction of Riemann surfaces with corona. J. Geom. Anal. [**8**]{} (1998), 341-347. L. Carleson, Interpolation of bounded analytic functions and the corona problem. Ann. of Math. [**76**]{} (1962), 547-559. T. W. Gamelin, Uniform algebras and Jensen measures. London Math. Soc. Lecture Notes Series [**32**]{}, Cambridge University Press, 1978. H. Grauert and R. Remmert, Theorie der Steinschen Räume. Springer-Verlag, Berlin, 1977. P. Jones and D. Marshall, Critical points of Green’s functions, harmonic measure and the corona theorem. Ark. Mat. [**23**]{} no.2 (1985), 281-314. F. Lárusson, Holomorphic functions of slow growth on nested covering spaces of compact manifolds. Canad. J. Math. [**52**]{} (5) (2000), 982-998. T. Napier, Convexity properties of coverings of smooth projective varieties. Math. Ann. [**286**]{} (1990), 433-479.

[^1]: Research supported in part by NSERC. 2000 [*Mathematics Subject Classification*]{}. Primary 32A10. Secondary 14E20. . Bounded holomorphic function, corona problem, transformation group.
