  Let $S^n(X)$ be the $n$-fold symmetric product of a compact connected Riemann
surface $X$ of genus $g$ and gonality $d$. We prove that $S^n(X)$ admits a
K\"ahler structure such that all the holomorphic bisectional curvatures are
nonpositive if and only if $n < d$. Let ${\mathcal Q}_X(r,n)$ be the Quot
scheme parametrizing the torsion quotients of ${\mathcal O}^{\oplus r}_X$ of
degree $n$. If $g \geq 2$ and $n \leq 2g-2$, we prove that ${\mathcal
Q}_X(r,n)$ does not admit a K\"ahler structure such that all the holomorphic
bisectional curvatures are nonnegative.
