The semiregularity theorem for equivariant noncommutative varieties

TL;DR

This paper extends the semiregularity theorem to equivariant noncommutative varieties, impacting derived categories and addressing questions related to the Hodge conjecture for abelian fourfolds.

Contribution

It generalizes a classical theorem to noncommutative geometry with group actions and shows invariant categories often have geometric origins.

Findings

Generalization of semiregularity theorem to noncommutative setting

Application to twisted derived categories and Hodge conjecture

Invariant categories under finite group actions are often geometric

Abstract

We generalize the classical semiregularity theorem of Buchweitz and Flenner to the setting of noncommutative algebraic geometry, with group actions. This applies in particular to twisted derived categories, in which case it answers a question of Markman and streamlines part of his proof of the Hodge conjecture for abelian fourfolds. Along the way, we prove that for many finite group actions on derived categories of varieties, the invariant category is of geometric origin.