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

Abstract

AbstractLet Y be an infinite covering space of a projective manifold M in N of dimension n ≥ 2. Let C be the intersection with M of at most n − 1 generic hypersurfaces of degree d in N. The preimage X of C in Y is a connected submanifold. Let φ be the smoothed distance from a fixed point in Y in a metric pulled up from M. Let φ(X) be the Hilbert space of holomorphic functions f on X such that f2e−φ is integrable on X, and define φ(Y) similarly. Our main result is that (under more general hypotheses than described here) the restriction φ(Y) → φ(X) is an isomorphism for d large enough.This yields new examples of Riemann surfaces and domains of holomorphy in n with corona. We consider the important special case when Y is the unit ball in 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 . Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from X to .