Sobolev Regularity for the Bergman Projection on Relatively Compact Domains in Hermitian manifolds

TL;DR

This paper establishes conditions under which the Bergman projection exhibits Sobolev regularity on certain domains within Hermitian manifolds, extending previous results and providing practical examples.

Contribution

It generalizes existing Sobolev regularity results for the Bergman projection to domains in Hermitian manifolds, including new examples in Hopf manifolds.

Findings

Sufficient conditions for Sobolev regularity of the Bergman projection.

Extension of regularity results to domains with Lipschitz boundary in Hermitian manifolds.

Application to domains in Hopf manifolds with suitable metrics.

Abstract

Generalizing a result of Berndtsson and Charpentier, we provide sufficient conditions for $L^2$ Sobolev regularity of the Bergman projection acting on $L^2$ sections of a holomorphic line bundle restricted to a relatively compact domain with Lipschitz boundary in a Hermitian manifold. We provide examples to show that our methods work for domains in Hopf manifolds endowed with a suitable Hermitian metric.