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Holomorphic Functions of Slow Growth on Nested Covering Spaces of Compact Manifolds


Published:20001001
Printed: Oct 2000
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Abstract
Let $Y$ be an infinite covering space of a projective manifold
$M$ in $\P^N$ of dimension $n\geq 2$. Let $C$ be the intersection with
$M$ of at most $n1$ generic hypersurfaces of degree $d$ in $\mathbb{P}^N$.
The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $\phi$
be the smoothed distance from a fixed point in $Y$ in a metric pulled up
from $M$. Let $\O_\phi(X)$ be the Hilbert space of holomorphic
functions $f$ on $X$ such that $f^2 e^{\phi}$ is integrable on $X$, and
define $\O_\phi(Y)$ similarly. Our main result is that (under more
general hypotheses than described here) the restriction $\O_\phi(Y)
\to \O_\phi(X)$ is an isomorphism for $d$ large enough.
This yields new examples of Riemann surfaces and domains of holomorphy
in $\C^n$ with corona. We consider the important special case when $Y$
is the unit ball $\B$ in $\C^n$, and show that for $d$ large enough,
every bounded holomorphic function on $X$ extends to a unique function
in the intersection of all the nontrivial weighted Bergman spaces on
$\B$. Finally, assuming that the covering group is arithmetic, we
establish three dichotomies concerning the extension of bounded
holomorphic and harmonic functions from $X$ to $\B$.
MSC Classifications: 
32A10, 14E20, 30F99, 32M15 show english descriptions
Holomorphic functions Coverings [See also 14H30] None of the above, but in this section Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15]
32A10  Holomorphic functions 14E20  Coverings [See also 14H30] 30F99  None of the above, but in this section 32M15  Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15]
