On the Kähler structures over Quot schemes, II

Abstract

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 Kähler metric whose holomorphic bisectional curvatures are all nonnegative.