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Mirror symmetry for abelian varieties

Vasily Golyshev·Valery Lunts·Dmitri Orlov
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Abstract

We work out the notion of mirror symmetry for abelian varieties and study its properties. Our construction are based on the correspondence between two $Q$--algebraic groups. One is the Hodge (or special Mumford--Tate) group. The second group $\bar{Spin(A)}$ is defined as follows: the group of autoequivalences of the bounded derived category of coherent sheaves acts on the total cohomology $H(A,Q)$ of an abelian variety $A$ via algebraic correspondences. The group $\bar{Spin(A)}$ is now the Zariski closure of its image in $GL(H(A,Q))$. Our constructions are compatible with the picture of mirror symmetry sketched by Kontsevich, Morrison, and others.

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Vasily Golyshev, Valery Lunts, Dmitri Orlov. 2009-11-29. Mirror symmetry for abelian varieties. https://arxiv.org/abs/math/9812003

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