Restriction of the Poincaré bundle to a Calabi-Yau hypersurface

Details Restriction of the Poincaré bundle to a Calabi-Yau hypersurface Restriction of the Poincaré bundle to a Calabi-Yau hypersurface

1999-02-25

arxiv.org/abs/math/9902145v1

Mathematics

Algebraic Geometry

AMSLaTex file. To appear in Crelles J

Scientific paper

Let $\cMx$ be the moduli space of stable vector bundles of rank $n\geq 3$ and determinant $\xi$ over a connected Riemann surface $X$, with $n$ and $d(\xi)$ coprime. Let $D$ be a Calabi-Yau hypersurface of $\cMx$. Denote by $U_D$ the restriction of the universal bundle to $X\times D$. It is shown that the restriction $(U_D)_x$ to $x\times D$ is stable, for any $x\in X$. Furthermore, for a general curve the connected component of the moduli space of semistable sheaves over $D$, containing $(U_D)_x$, is isomorphic to $X$. It is also shown that $U_D$ is stable for any polarisation, and the connected component of the moduli space of semistable sheaves over $X\times D$, containing $U_D$, is isomorphic to the Jacobian. Moreover, this is an isomorphism of polarised varieties, and hence such a moduli spaces determine the Reimann surface.

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