A note on co-Higgs bundles

TL;DR

This paper explores the conditions under which co-Higgs bundles are semistable on certain complex projective varieties and investigates criteria for nilpotent co-Higgs structures on surfaces.

Contribution

It establishes a link between the semistability of co-Higgs bundles and the properties of the underlying variety, and provides criteria for nilpotency of rank two co-Higgs structures.

Findings

Semistability of co-Higgs bundles implies semistability of bundles on varieties with nonnegative Kodaira dimension.

Criteria for surfaces to have vanishing global sections of tangent and symmetric powers, leading to nilpotent co-Higgs structures.

Conditions under which rank two co-Higgs structures are necessarily nilpotent.

Abstract

We show that for any ample line bundle on a smooth complex projective variety with nonnegative Kodaira dimension, the semistability of co-Higgs bundles of implies the semistability of bundles. Then we investigate the criterion for surface $X$ to have $H^0(T_X)=H^0(S^2T_X)=0$, which implies that any co-Higgs structure of rank two is nilpotent.