Mirror symmetry for abelian varieties

TL;DR

This paper develops a notion of mirror symmetry specifically for abelian varieties, linking algebraic groups and derived categories to deepen understanding of their geometric and categorical dualities.

Contribution

It introduces a new framework connecting the Hodge group and a spin group via algebraic correspondences, aligning with existing mirror symmetry theories.

Findings

Established a correspondence between Hodge and spin groups for abelian varieties.

Constructed the group ar{Spin(A)} as the Zariski closure of autoequivalence actions.

Demonstrated compatibility with Kontsevich and Morrison's mirror symmetry models.

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.