**
Alejandra
Vicente
Colmenares,
Pure
Mathematics,
University
of
Waterloo**

*
"Semistable
rank
2
co-Higgs
bundles
over
Hirzebruch
surfaces"*

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
$\MPP(c_1,c_2)$.