Automorphisms of the Quot schemes associated to compact Riemann surfaces

TL;DR

This paper investigates the automorphism group of Quot schemes associated with compact Riemann surfaces, showing that the projective linear group forms the connected component of automorphisms and that the surface is uniquely determined by the Quot scheme.

Contribution

It establishes that the automorphism group of the Quot scheme has a specific structure and that the Riemann surface can be recovered from the Quot scheme's isomorphism class.

Findings

PGL(r,C) is the connected component of Aut(Q) containing the identity.

The isomorphism class of Q determines the Riemann surface X uniquely.

Automorphisms of Q are closely related to the geometry of X.

Abstract

Let X be a compact connected Riemann surface of genus at least two. Fix positive integers r and d. Let Q denote the Quot scheme that parametrizes the torsion quotients of {\mathcal O}^{\oplus r}_X of degree d. This Q is also the moduli space of vortices for the standard action of U(r) on {\mathbb C}^r. The group \text{PGL}(r, {\mathbb C}) acts on Q via the action of $\text{GL}(r, {\mathbb C})$ on ${\mathcal O}^{\oplus r}_X$. We prove that this subgroup $\text{PGL}(r, {\mathbb C})$ is the connected component, containing the identity element, of the holomorphic automorphism group Aut(\mathcal Q). As an application of it, we prove that the isomorphism class of the complex manifold Q uniquely determines the isomorphism class of the Riemann surface X.