Automorphisms of the Quot schemes associated to compact Riemann surfaces

2012/11/14 by Biswas, Indranil, Dhillon, Ajneet, Hurtubise, Jacques
#14H60 #14J50 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1211.3306

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^{⊕ 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 PGL(r, \mathbb C) acts on Q via the action of GL(r, \mathbb C) on \mathcal O^{⊕ r}_X. We prove that this subgroup 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.

Related

• Automorphisms of the generalized quot schemes
• A generalized Quot scheme and meromorphic vortices
• Quot schemes and Ricci semipositivity
• Automorphisms of relative Quot Schemes
• Quot scheme and deformation quantization