Hodge theory of twisted derived categories and the period-index problem

TL;DR

This paper develops a Hodge-theoretic framework for twisted derived categories, linking it to the period-index problem, and applies it to construct Hodge classes, counterexamples, and prove conjectures.

Contribution

It introduces twisted Mukai structures for Brauer classes on smooth varieties, advancing the understanding of the period-index problem and Hodge theory.

Findings

Constructed Hodge classes implying period-index bounds

Provided counterexamples to the integral Hodge conjecture on Severi-Brauer varieties

Proved the integral Hodge conjecture for derived categories of Deligne-Mumford surfaces

Abstract

We study the Hodge theory of twisted derived categories and its relation to the period-index problem. Our main contribution is the development of a theory of twisted Mukai structures for topologically trivial Brauer classes on arbitrary smooth proper varieties and in families. As applications, we construct Hodge classes whose algebraicity would imply period-index bounds; construct new counterexamples to the integral Hodge conjecture on Severi-Brauer varieties; and prove the integral Hodge conjecture for derived categories of Deligne-Mumford surfaces.