We use the framework of Quot schemes to give a novel description of the
moduli spaces of stable n-pairs, also interpreted as gauged vortices on a
closed Riemann surface with target Mat(r x n, C), where n >= r. We then show
that these moduli spaces embed canonically into certain Grassmann manifolds,
and thus obtain natural Kaehler metrics of Fubini-Study type; these spaces are
smooth at least in the local case r=n.

Let ${\mathcal P}{\mathcal M}^\alpha_s$ be a moduli space of stable parabolic
vector bundles of rank $n \geq 2$ and fixed determinant of degree $d$ over a
compact connected Riemann surface $X$ of genus $g(X) \geq 2$. If $g(X) = 2$,
then we assume that $n > 2$. Let $m$ denote the greatest common divisor of $d$,
$n$ and the dimensions of all the successive quotients of the quasi-parabolic
filtrations. We prove that the cohomological Brauer group ${\rm Br}({\mathcal
P}{\mathcal M}^\alpha_s)$ is isomorphic to the cyclic group ${\mathbb Z}/
m{\mathbb Z}$.

Fix a $C^\infty$ principal $G$--bundle $E^0_G$ on a compact connected Riemann
surface $X$, where $G$ is a connected complex reductive linear algebraic group.
We consider the gradient flow of the Yang--Mills--Higgs functional on the
cotangent bundle of the space of all smooth connections on $E^0_G$. We prove
that this flow preserves the subset of Higgs $G$--bundles, and, furthermore,
the flow emanating from any point of this subset has a limit. Given a Higgs
$G$--bundle, we identify the limit point of the integral curve passing through
it.

Let $X$ be a compact Riemann surface $X$ of genus at--least two. Fix a
holomorphic line bundle $L$ over $X$. Let $\mathcal M$ be the moduli space of
Hitchin pairs $(E ,\phi\in H^0(End(E)\otimes L))$ over $X$ of rank $r$ and
fixed determinant of degree $d$. We prove that, for some numerical conditions,
$\mathcal M$ is irreducible, and that the isomorphism class of the variety
$\mathcal M$ uniquely determines the isomorphism class of the Riemann surface
$X$.

Let (X, \omega) be a compact connected Kaehler manifold of complex dimension
d and E_G a holomorphic principal G-bundle on X, where G is a connected
reductive linear algebraic group defined over C. Let Z (G) denote the center of
G.