Automorphisms of the Quot schemes associated to compact Riemann surfaces

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.