  In the present thesis we study the geometry of the moduli spaces of
Bradlow-Higgs triples on a smooth projective curve $C$. $(E,\varphi, s)$ is a
Bradlow-Higgs triple if $(E,\varphi)$ is a Higgs bundle and $s$ is a non-zero
global section of $E$. There is a family of stability conditions for triples
that depends on a positive real parameter $\sigma$. The moduli spaces
$\mathcal{M}_\sigma^{r,d}$ of $\sigma$-semistable triples of rank $r$ and
degree $d$ vary with $\sigma$. The phenomenon arising from this is known as
wall-crossing. In the first half of the thesis we will examine how the moduli
spaces $\mathcal{M}_\sigma^{r,d}$ and their universal additive invariants
change as $\sigma$ varies, for the case $r=2$. In particular we will study the
case of $\sigma$ very close to 0, for which $\mathcal{M}_\sigma^{r,d}$ relates
to the moduli space of stable Higgs bundles, and $\sigma$ very large, for which
$\mathcal{M}_\sigma^{r,d}$ is a relative Hilbert scheme of points for the
family of spectral curves. Some of these results will be generalized to
Bradlow-Higgs triples with poles. In the second half we will prove a formula
relating the cohomology of $\mathcal{M}_\sigma^{2,d}$ for small $\sigma$ and
$d$ odd and the perverse filtration on the cohomology of the moduli space of
stable Higgs bundles. The formula is not far from the generalized Macdonald
formulas found in the works of Migliorini-Shende, Maulik-Yun and
Migliorini-Shende-Viviani. We will also partially generalize this result to the
case of rank greater than 2.
