Infinitesimal deformations of the tangent bundle of a moduli space of vector bundles over a curve

Abstract

Fix a line bundle $\xi$ on a connected smooth complex projective curve $X$ of genus at least three. Let $\mathcal{N}$ denote the moduli space of all stable vector bundles over $X$ of rank $n$ and determinant $\xi$. We assume that $n\geq 3$ and coprime to $\operatorname{degree}(\xi)$; If $\operatorname{genus}(X)\leq 4$, then we also assume that $n \geq 4$. We prove that $H^i(\mathcal{N}, \End(T\mathcal{N}\mkern2mu)) = H^i(X, \mathcal{O}_X)$ for $i= 0,1$.