# Quot schemes and Ricci semipositivity

## TL;DR

This paper investigates the geometric properties of a moduli space of torsion quotients on a Riemann surface, showing it cannot support a Kähler metric with semipositive Ricci curvature due to its anticanonical bundle.

## Contribution

It proves that the anticanonical line bundle of the quot scheme ${
m extbf{Q}}_X(r,d)$ is not nef, implying the space cannot have a Ricci semipositive Kähler metric.

## Key findings

- Anticanonical bundle of ${
m extbf{Q}}_X(r,d)$ is not nef
- ${
m extbf{Q}}_X(r,d)$ admits no Kähler metric with semipositive Ricci curvature
- The result links geometric properties of the quot scheme to curvature conditions

## Abstract

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${\mathcal Q}_X(r,d)$ be the quot scheme that parametrizes all the torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. This ${\mathcal Q}_X(r,d)$ is also a moduli space of vortices on $X$. Its geometric properties have been extensively studied. Here we prove that the anticanonical line bundle of ${\mathcal Q}_X(r,d)$ is not nef. Equivalently, ${\mathcal Q}_X(r,d)$ does not admit any K\"ahler metric whose Ricci curvature is semipositive.