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

TL;DR

This paper explores Lagrangian-invariant objects in the derived category of an abelian variety, constructing a phase function related to stability conditions, and relates these concepts to mirror symmetry and the structure of the stability space.

Contribution

It introduces a natural phase function for LI-objects associated with the complexified ample cone, linking endofunctors, group extensions, and mirror symmetry in the context of abelian varieties.

Findings

Constructed a phase function on LI-objects for each element of the complexified ample cone.

Linked the phase function to Bridgeland stability in the case of abelian surfaces.

Showed the stability space component contains all full stability conditions for abelian surfaces.

Abstract

We continue the study of Lagrangian-invariant objects (LI-objects for short) in the derived category $D^b(A)$ of coherent sheaves on an abelian variety, initiated in arXiv:1109.0527. For every element of the complexified ample cone $D_A$ 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 (see math.AG/0307164). The construction is based on the relation between endofunctors of $D^b(A)$ 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 byproduct of our study of LI-objects we show that the Bridgeland's component of the stability space of an abelian surface contains all full stabilities.