Holomorphic vector bundles on non-algebraic surfaces

TL;DR

This paper addresses the existence of holomorphic vector bundles on non-algebraic surfaces, providing solutions for rank 2 bundles on K3 surfaces and arbitrary rank on class VII surfaces using Donaldson and deformation theories.

Contribution

It offers new existence results for holomorphic vector bundles on specific non-algebraic surfaces, expanding the understanding of their structure and classification.

Findings

Existence of rank 2 bundles on K3 surfaces established.

Existence of arbitrary rank bundles on class VII surfaces demonstrated.

Methods can be extended to other non-algebraic surfaces.

Abstract

The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces.