The period-index conjecture for abelian threefolds and Donaldson-Thomas theory

TL;DR

This paper proves the period-index conjecture for unramified Brauer classes on abelian threefolds by developing a new theory of Donaldson-Thomas invariants in Calabi-Yau categories, linking algebraic and Hodge-theoretic properties.

Contribution

It introduces a novel framework of reduced Donaldson-Thomas invariants for 3-Calabi-Yau categories and applies it to resolve the period-index conjecture for abelian threefolds.

Findings

Proved the period-index conjecture for unramified Brauer classes on abelian threefolds.

Established the algebraicity of certain Hodge classes on twisted derived categories.

Deduced the integral Hodge conjecture for generically twisted abelian threefolds.

Abstract

We prove the period-index conjecture for unramified Brauer classes on abelian threefolds. To do so, we develop a theory of reduced Donaldson-Thomas invariants for 3-dimensional Calabi-Yau categories, with the feature that the noncommutative variational integral Hodge conjecture holds for classes with nonvanishing invariant. The period-index result is then proved by interpreting it as the algebraicity of a Hodge class on the twisted derived category, and specializing within the Hodge locus to an untwisted abelian threefold with nonvanishing invariant. As a consequence, we also deduce the integral Hodge conjecture for generically twisted abelian threefolds.