PhD Thesis Defence

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, we study the geometry of these objects (in the rank 2 case) over Hirzebruch surfaces, giving special emphasis to $\P \times \P$. Two main topics can be identified throughout the dissertation: non-emptiness of the moduli spaces of rank 2 semistable co-Higgs bundles over Hirzebruch surfaces, and the description of these moduli spaces over $\P \times \P$.

The existence problem consists in determining for which pairs of Chern classes $(c_1,c_2)$ there exists a non-trivial semistable rank 2 co-Higgs bundle with Chern classes $c_1$ and $c_2$. We approach this problem from two different perspectives. On one hand, we restrict ourselves to certain natural choices of $c_1$ and give necessary and sufficient conditions on $c_2$ that guarantee the existence of non-trivial semistable co-Higgs bundles with these Chern classes; we do this for arbitrary polarizations when $c_2 \leq 2$. On the other hand, for arbitrary $c_1$, we also provide necessary and sufficient conditions on $c_2$ that ensure the existence of non-trivial semistable co-Higgs bundles; however, we only do this for the standard polarization.

As for the description of the moduli spaces $\MPP(c_1,c_2)$ of rank 2 semistable co-Higgs bundles over $\P \times \P$, we restrict ourselves to the standard polarization. We then discuss how to use the spectral construction and the Hitchin correspondence to understand generic rank 2 semistable co-Higgs bundles. Furthermore, we give an explicit description of the moduli spaces when $c_2=0, 1$ for certain choices of $c_1$. Finally, we explore the first order deformations of points in the moduli space $\MPP(c_1,c_2)$.