On the K\"ahler structures over Quot schemes, II

TL;DR

This paper proves that the Quot scheme parametrizing torsion quotients over a compact Riemann surface of genus at least 2 cannot admit a Kähler metric with nonnegative holomorphic bisectional curvature.

Contribution

It establishes a nonexistence result for Kähler metrics with nonnegative bisectional curvature on certain Quot schemes over higher genus surfaces.

Findings

Quot scheme ${ m Q}(r,d)$ does not admit such Kähler metrics.

The result applies to all positive integers r and d.

The proof involves curvature and geometric analysis techniques.

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 K\"ahler metric whose holomorphic bisectional curvatures are all nonnegative.