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Let  X  be a compact connected Riemann surface of genus at least 2. Fix positive integers  r  and  d . Let   $\mathcal {Q}$   denote the Quot scheme that parameterizes all the torsion quotients of   ${\mathcal O}^{\oplus r}_X$   of degree  d . This   $\mathcal {Q}$   is also the moduli space of vortices for the standard action of  U ( r ) on   ${\mathbb C}^r$  . The group   $\hbox {PGL}(r, {\mathbb C})$   acts faithfully on   $\mathcal {Q}$   via the standard action of   $\hbox {GL}(r, {\mathbb C})$   on   ${\mathcal O}^{\oplus r}_X$  . We prove that this subgroup   $\hbox {PGL}(r, {\mathbb C})$   is the connected component, containing the identity element, of the holomorphic automorphism group   $\hbox {Aut}(\mathcal {Q})$  . As an application, we prove that the isomorphism class of the complex manifold   $\mathcal {Q}$   uniquely determines the isomorphism class of the Riemann surface  X .

Automorphisms of the Quot Schemes Associated to Compact Riemann Surfaces