CY Manifolds with Locally Symmetric Moduli Spaces

Details CY Manifolds with Locally Symmetric Moduli Spaces CY Manifolds with Locally Symmetric Moduli Spaces

2008-06-25

arxiv.org/abs/0806.4010v4

Mathematics

Algebraic Geometry

one reference is added and several typoes are corrected. Some changes are made in the introduction

Scientific paper

There is a natural sequence of CY manifolds that are double covers of the projective g dimensional spaces ramified over 2g+2 hyperplanes. We observe that some of them are obtained as quotient of the action of the semi-direct product of g-1 copies of cyclic Z/2Z groups with the symmetric g group on the product of g-copies of hyper-elliptic curves of genus g. The quotient is a double cover of the projective g space ramified over 2g+2 hyperplanes. This construction generalizes the construction of a Kummer surface. The Kodaira-Spencer classes on the Jacobian are invariant under the action of the group. Thus they form a basis of Kodaira-Spencer classes on the CY manifold. Since the bracket of any Kodaira-Spencer classes on the Jacobian are zero, then they will be zero on the CY manifold. This implies that the moduli space of those CY manifolds is a locally symmetric space.

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