Shifting numbers of abelian varieties via bounded t-structures

TL;DR

This paper proves that shifting numbers, which measure how autoequivalences translate objects in triangulated categories, form a quasimorphism on the autoequivalence group of abelian varieties of any dimension, generalizing previous results.

Contribution

It establishes that shifting numbers define a quasimorphism for autoequivalences of derived categories of abelian varieties, extending prior work from elliptic curves and surfaces.

Findings

Shifting numbers are quasimorphisms on autoequivalence groups of abelian varieties.

An alternative definition of shifting numbers via bounded t-structures is provided.

The full Bridgeland stability condition is not necessary for computing shifting numbers.

Abstract

The shifting numbers measure the asymptotic amount by which an endofunctor of a triangulated category translates inside the category, and are analogous to Poincare translation numbers that are widely used in dynamical systems. Motivated by this analogy, Fan-Filip raised the following question: ``Do the shifting numbers define a quasimorphism on the group of autoequivalences of a triangulated category?" An affirmative answer was given by Fan-Filip for the bounded derived category of coherent sheaves on an elliptic curve or an abelian surface, via properties of the spaces of Bridgeland stability conditions on these categories. We prove in this article that the question has an affirmative answer for abelian varieties of arbitrary dimensions, generalizing the result of Fan-Filip. One of the key steps is to establish an alternative definition of the shifting numbers via bounded t-structures on triangulated categories. In particular, the full package of a Bridgeland stability condition (a bounded t-structure, and a central charge on a charge lattice) is not necessary for the purpose of computing the shifting numbers.