Infinitesimal deformations of some Quot schemes

Let \(E\) be a vector bundle on a smooth complex projective curve \(C\) of genus at least two. Let \(\mathcal{Q}(E,d)\) be the Quot scheme parameterizing the torsion quotients of \(E\) of degree \(d\). We compute the cohomologies of the tangent bundle \(T_{\mathcal{Q}(E,d)}\). In particular, the space of infinitesimal deformations of \(\mathcal{Q}(E,d)\) is computed. Kempf and Fantechi computed the space of infinitesimal deformations of \(\mathcal{Q}(\mathcal{O}_C,d)\,=\, C^{(d)}\). We also explicitly describe the infinitesimal deformations of \(\mathcal{Q}(E,d)\).