Twisted holomorphic symplectic forms

TL;DR

This paper characterizes compact Kähler manifolds with nondegenerate holomorphic 2-forms valued in line bundles, showing they are finite cyclic covers of hyperkähler manifolds and analyzing their fundamental groups.

Contribution

It establishes a classification of such manifolds as finite cyclic covers of hyperkähler manifolds and describes their fundamental group structure.

Findings

Manifolds are finite cyclic covers of hyperkähler manifolds

Holomorphic 2-forms are parallel with respect to the induced connection

Fundamental groups have specific structural properties

Abstract

We show that a compact Kahler manifold admitting a nondegenerate holomorphic 2-form valued in a line bundle is a finite cyclic cover of a hyperkahler manifold. With respect to the connection induced by the locally hyperkahler metric, the form is parallel. We then describe the structure of the fundamenal group of such manifolds and derive some consequences.