On the Kähler structures over Quot schemes

Abstract

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 Kähler 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 Kähler structure such that all the holomorphic bisectional curvatures are nonnegative.