Vanishing theorems for parabolic Higgs bundles

TL;DR

This paper proves a Kodaira vanishing theorem for semistable parabolic Higgs bundles with trivial parabolic Chern classes in characteristic zero, leading to a semipositivity result and a vanishing theorem for complex variations of Hodge structure.

Contribution

It establishes a new Kodaira vanishing theorem for parabolic Higgs bundles in characteristic zero, extending previous positive characteristic results.

Findings

Proves Kodaira vanishing for semistable parabolic Higgs bundles with trivial parabolic Chern classes.

Derives a semipositivity theorem from the vanishing result.

Establishes a Kodaira-Saito vanishing theorem for complex variations of Hodge structure.

Abstract

This is a sequel to "Kodaira-Saito vanishing via Higgs bundles in positive characteristic" (arXiv:1611.09880). However, unlike the previous paper, all the arguments here are in characteristic zero. The main result is a Kodaira vanishing theorem for semistable parabolic Higgs bundles with trivial parabolic Chern classes. This implies a general semipositivity theorem. This also implies a Kodaira-Saito vanishing theorem for complex variations of Hodge structure.