2-Nilpotent co-Higgs structures

Abstract

A co-Higgs sheaf on a smooth complex projective variety X 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, especially over projective spaces.