Kodaira dimension and zeros of holomorphic one-forms, revisited

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.