On the moduli spaces of semistable co-Higgs bundles over Hirzebruch surfaces

Abstract: It has been observed by S. Rayan that the complex projective surfaces that potentially admit non-trivial examples of semistable co-Higgs bundles must be found at the lower end of the Enriques-Kodaira classification. Motivated by this remark, it is natural to study the geometry of these objects over Hirzebruch surfaces.

In this talk, we present necessary and sufficient conditions on the Chern classes $c_1, c_2$ of a bundle that guarantee the non-emptiness of the moduli space $\mathcal{M}^\mathrm{co}(c_1,c_2)$ or rank 2 semistable co-Higgs bundles over the Hirzebruch surface $\mathbb{F}_n$; we do this with respect to the standard polarization. In the case of $\mathbb{F}_0 = \mathbb{P}^1 \times \mathbb{P}^1$, we give an explicit description of the moduli spaces for certain choices of $c_1$ and $c_2$.