2-Nilpotent co-Higgs structures

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.