Infinitesimal deformations of a Calabi-Yau hypersurface of the moduli space of stable vector bundles over a curve

Abstract

Let $X$ be a compact connected Riemann surface of genus $g$, with $g\ge 2$, and ${\mathcal{M}}_{\xi }$ a smooth moduli space of fixed determinant semistable vector bundles of rank $n$, with $n\ge 2$, over $X$. Take a smooth anticanonical divisor $D$ on ${\mathcal{M}}_{\xi }$. So $D$ is a Calabi-Yau variety. We compute the number of moduli of $D$, namely $\dim H^1(D,T_D)$, to be $3g-4+\dim H^0({\mathcal{M}}_{\xi },K_{{\mathcal{M}}_{\xi }}^{-1})$. Denote by $\mathcal{N}$ the moduli space of all such pairs $(X^{\prime },D^{\prime })$, namely $D^{\prime }$ is a smooth anticanonical divisor on a smooth moduli space of semistable vector bundles over the Riemann surface $X^{\prime }$. It turns out that the Kodaira-Spencer map from the tangent space to $\mathcal{N}$, at the point represented by the pair $(X,D)$, to $H^1(D,T_D)$ is an isomorphism. This is proved under the assumption that if $g=2$, then $n\ne 2,3$, and if $g=3$, then $n\ne 2$.

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