Phases of Lagrangian-invariant objects in the derived category of an abelian variety

Abstract

In this paper we study Lagrangian-invariant objects (LI objects for short) in the derived category Db (A) of coherent sheaves on an abelian variety. For every element of the complexified ample cone DA we construct a natural phase function on the set of LI objects, which in the case dim A=2 gives the phases with respect to the corresponding Bridgeland stability. The construction is based on the relation between endofunctors of Db (A) and a certain natural central extension of groups, associated with D A viewed as a Hermitian symmetric space. In the case when A is a power of an elliptic curve, we show that our phase function has a natural interpretation in terms of the Fukaya category of the mirror dual abelian variety. As a by-product of our study of LI objects we show that the Bridgeland component of the stability space of an abelian surface contains all full stabilities. © 2014 by Kyoto University.