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18 January 2022
  » arxiv » math.AG/9904033

 Article overview

Infinitesimal deformations of a Calabi-Yau hypersurface of the moduli space of stable vector bundles over a curve
Indranil Biswas ; Leticia Brambila-Paz ;
Date 8 Apr 1999
Subject Algebraic Geometry | math.AG
AbstractLet $X$ be a compact connected Riemann surface of genus $g$, with $ggeq 2$, and ${cal M}_{xi}$ a smooth moduli space of fixed determinant semistable vector bundles of rank $n$, with $ngeq 2$, over $X$. Take a smooth anticanonical divisor $D$ on ${cal M}_{xi}$. So $D$ is a Calabi-Yau variety. We compute the number of moduli of $D$, namely $dim H^1(D, T_D)$, to be $3g-4 + dim H^0({cal M}_{xi}, K^{-1}_{{cal M}_{xi}})$. Denote by $cal N$ the moduli space of all such pairs $(X’,D’)$, namely $D’$ is a smooth anticanonical divisor on a smooth moduli space of semistable vector bundles over the Riemann surface $X’$. It turns out that the Kodaira-Spencer map from the tangent space to $cal N$, 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 $n eq 2,3$, and if $g=3$, then $n eq 2$.
Source arXiv, math.AG/9904033
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