On vanishing theorems for Higgs bundles

TL;DR

This paper extends vanishing theorems for Hermitian Higgs bundles from Kähler to general compact complex manifolds, establishing new results on invariant sections and curvature conditions.

Contribution

It generalizes key vanishing theorems for Hermitian Higgs bundles to compact complex manifolds and explores conditions for the existence of invariant sections.

Findings

Vanishing theorems hold on compact complex manifolds, not just Kähler.

Invariant sections vanish under negativity of the Hitchin-Simpson mean curvature.

Invariant sections of certain Higgs bundles are parallel when the bundle is weak Hermitian-Yang-Mills.

Abstract

We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the base manifold was K\"ahler, also holds when the manifold is compact complex. From this fact and some basic properties of Hermitian Higgs bundles, we conclude several results. In particular we show that, in analogy to the classical case, there are vanishing theorems for invariant sections of tensor products of Higgs bundles. Then, we prove that a Higgs bundle admits no nonzero invariant sections if there is a condition of negativity on the greatest eigenvalue of the Hitchin-Simpson mean curvature. Finally, we prove that invariant sections of certain tensor products of a weak Hermitian-Yang-Mills Higgs bundle are all parallel in the classical sense.