Kodaira dimension and zeros of holomorphic one-forms

TL;DR

This paper proves that on a smooth complex projective variety of general type, every holomorphic one-form must have at least one zero, using advanced techniques from generic vanishing theory.

Contribution

It establishes a new vanishing property of holomorphic one-forms on varieties of general type, linking geometric properties with Hodge theory.

Findings

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

Uses generic vanishing theory for Hodge D-modules.

Provides new insights into the structure of varieties of general type.

Abstract

We show that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point. The proof uses generic vanishing theory for Hodge D-modules on abelian varieties.