|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^1 x P^1. 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^1 x P^1.
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 ≤ 2. On the other hand, for arbitrary
c_1, we also provide necessary and sufficient conditions on c_2 that ensure the existence of nontrivial semistable co-Higgs bundles; however, we only do this for the standard polarization.
As for the description of the moduli spaces M(c_1; c_2) of rank 2 semistable co-Higgs
bundles over P^1 x P^1, 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 M(c_1; c_2).