$2$-nilpotent co-Higgs structures

TL;DR

This paper constructs specific 2-nilpotent co-Higgs sheaves on certain rational surfaces and projective spaces, and investigates their existence and non-existence conditions.

Contribution

It introduces new constructions of 2-nilpotent co-Higgs sheaves on rational surfaces and projective spaces using Hartshorne-Serre correspondence.

Findings

Constructed rank two co-Higgs sheaves on rational surfaces

Built rank three co-Higgs sheaves on ^3

Analyzed non-existence conditions over projective spaces

Abstract

A co-Higgs sheaf is a pair of a torsion-free coherent sheaf $\mathcal{E}$ and a global section of $\mathcal{E}nd(\mathcal{E})\otimes T_X$ with $T_X$ the tangent bundle. We construct $2$-nilpotent co-Higgs sheaves of rank two for some rational surfaces and of rank three for $\mathbb{P}^3$, using the Hartshorne-Serre correspondence. Then we investigate the non-existence, specially over projective spaces.