Kodaira dimension and zeros of holomorphic one-forms, revisited

TL;DR

This paper provides a new proof that holomorphic one-forms on general type complex projective varieties must vanish somewhere, using Simpson's results on Higgs bundles and cohomology jump loci.

Contribution

It offers an alternative proof to a known result, connecting Higgs bundles and cohomology loci, expanding the understanding of holomorphic forms on complex varieties.

Findings

Holomorphic one-forms on general type varieties must vanish somewhere.

New proof uses Simpson's relation between Higgs bundles and local systems.

Cohomology jump loci structure plays a key role.

Abstract

We give a new proof that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point, first proven by Popa and Schnell using generic vanishing theorems for Hodge modules. Our proof relies on Simpson's results on the relation between rank one Higgs bundles and local systems of one-dimensional complex vectors spaces, and the structure of the cohomology jump loci in their moduli spaces.