×

Mirror symmetry for abelian varieties. (English) Zbl 1014.14020

In the present paper, the authors define the relation of mirror symmetry on the class of pairs \((A,\omega_A)\), where \(A\) is an abelian variety and \(\omega_A\) is an element of the complexified ample cone \(C_A \subset NS_A (\mathbb{C})\) of the abelian variety \(A\). This notion of mirror symmetry is purely algebraic, on the one hand, but it is compatible with the usual analytic notion of mirror symmetry for Calabi-Yau manifolds, on the other hand.
As this notion of mirror symmetry for the so-called algebraic pairs \((A, \omega_A)\) of structured abelian varieties is extremely subtle and conceptually involved, it takes the authors ten chapters to rigorously establish it, via various delicate and original constructions, and to analyze its properties as well as its significance in the general realm of mirror symmetry.
After a very careful and motivating introduction to the subject of study, the first eight chapters provide the methodical and technical base for the main goal of the paper. Finally, in the remaining two chapters, the authors define the notion of mirror symmetry for algebraic pairs of complex tori and abelian varieties, analyze its properties and discuss another variant of this construction.
Without any doubt, this paper provides very substantial ideas and constructions towards the just as deep as fascinating problem of mirror symmetry in geometry and physics. Although being technically highly involved and methodically rather extensive (and intensive), the exposition is throughout very clear, detailed, rigorous and well-structured. The numerous remarks and comments that point to related developments, in this context, testify to both the comprising reflection of the authors and the actual value of this work.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14K99 Abelian varieties and schemes
17B45 Lie algebras of linear algebraic groups
PDFBibTeX XMLCite
Full Text: arXiv