Rank two nilpotent co-Higgs sheaves on complex surfaces

TL;DR

This paper classifies rank two nilpotent co-Higgs sheaves on complex surfaces, showing that semi-stable cases are constrained to uniruled surfaces, tori, or elliptic surfaces, up to finite étale cover.

Contribution

It provides a classification of semi-stable rank two nilpotent co-Higgs sheaves on complex surfaces, identifying the geometric types of surfaces where they can exist.

Findings

If semi-stable, the surface is uniruled, a torus, or a properly elliptic surface.

On tori and elliptic surfaces, the sheaves are strictly semi-stable.

The classification holds up to finite étale cover.

Abstract

Let $(\mathcal{E}, \phi)$ be a rank two co-Higgs vector bundles on a K\"ahler compact surface $X$ with $\phi\in H^0(X,End(\mathcal{E})\otimes T_X)$ nilpotent. If $(\mathcal{E}, \phi)$ is semi-stable, then one of the following holds up to finite \' etale cover: $i)$ $X$ is uniruled. $ii)$ $X$ is a torus and $(\mathcal{E}, \phi)$ is strictly semi-stable. $iii)$ $X$ is a properly elliptic surface and $(\mathcal{E}, \phi)$ is strictly semi-stable.