On the Kähler structures over Quot schemes, II

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.

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