  Let $X$ be a compact connected Riemann surface of genus $g$, with $g \geq 2$,
and let ${\mathcal O}_X$ denote the sheaf of holomorphic functions on $X$. Fix
positive integers $r$ and $d$ and let ${\mathcal Q}(r,d)$ be the Quot scheme
parametrizing all torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of
degree $d$. We prove that ${\mathcal Q}(r,d)$ does not admit a K\"ahler metric
whose holomorphic bisectional curvatures are all nonnegative.
