Mirror symmetry for abelian varieties

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 $\overline{Spin(A)}$ 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 $\overline{Spin(A)}$ 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.

LaTeX, 40 pages, no figures

Cited by in corpus (9)

• Remarks on A-branes, Mirror Symmetry, and the Fukaya category
• Lectures on Mirror Symmetry, Derived Categories, and D-branes
• Mirror symmetry and deformation quantization
• Non-commutative compactifications and elliptic curves
• Complex multiplication, rationality and mirror symmetry for abelian varieties
• Quantization and ``theta functions''
• Derived equivalences of Calabi-Yau fibrations
• On Brill-Noether loci over Quot schemes and a Torelli theorem
• Compactification of the moduli of polarized abelian varieties and mirror symmetry