1998/08/10 by Finnur Lárusson, Finnur Larusson, Larusson, Finnur
Mathematics · #14E20 #30F99 #32A10 (primary) #32M15 (secondary) #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Meromorphic and Entire Functions #math.AG #math.CV #msc:14E20 #msc:30F99 #msc:32A10 #msc:32M15
paper · pdf · doi:10.48550/arxiv.math/9808042
openalex publication_date 1998/08/10 · arxiv created 1999/03/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Y be an infinite covering space of a projective manifold M in PN of dimension n geq 2. Let C be the intersection with M of at most n-1 generic hypersurfaces of degree d in PN. 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 Ophi(X) be the Hilbert space of holomorphic functions f on X such that f2 e^(-phi) is integrable on X, and define Ophi(Y) similarly. Our main result is that (under more general hypotheses than described here) the restriction Ophi(Y) to Ophi(X) is an isomorphism for d large enough. This yields new examples of Riemann surfaces and domains of holomorphy in Cn with corona. We consider the important special case when Y is the unit ball B in Cn, 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.