  Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincar\'e
metric. For any integer $k\geq 2$, we investigate the Bergman kernel associated
to the holomorphic Hermitian line bundle $\Omega^{\otimes k}_X$, where $\O$ is
the holomorphic cotangent bundle of $X$. Our first main result estimates the
corresponding Bergman metric on $X$ in terms of the Poincar\'e metric. We then
consider a certain natural embedding of the symmetric product of $X$ into a
Grassmannian parametrizing subspaces of fixed dimension of the space of all
global holomorphic sections of $\Omega^{\otimes k}_X$. The Fubini-Study metric
on the Grassmannian restricts to a K\"ahler metric on the symmetric product of
$X$. The volume form for this restricted metric on the symmetric product is
estimated in terms of the Bergman kernel of $\Omega^{\otimes k}_X$ and the
volume form for the orbifold K\"ahler form on the symmetric product given by
the Poincar\'e metric on $X$.
