Weak Holomorphic Structures over K\"ahler Surfaces

TL;DR

This paper proves that certain Sobolev connections with (1,1) curvature on Hermitian bundles over K"ahler surfaces induce smooth holomorphic structures and can be approximated by smooth connections.

Contribution

It establishes the smoothness and approximation properties of Sobolev connections with (1,1) curvature on Hermitian bundles over K"ahler surfaces.

Findings

Sobolev $W^{1,2}$ connections with (1,1) curvature induce smooth holomorphic structures.

Such connections can be strongly approximated by smooth connections in $W^{1,p}$ norms for $p<2$.

The results extend the understanding of the regularity of connections on K"ahler surfaces.

Abstract

In this work we prove that any unitary Sobolev $W^{1,2}$ connection of an Hermitian bundle over a 2-dimensional K\"ahler manifold whose curvature is $(1,1)$ defines a smooth holomorphic structure. We prove moreover that such a connection can be strongly approximated in any $W^{1,p}$ ($p<2$) norm by smooth connections satisfying the same integrability condition.