Serre-Godeaux varieties and the etale index

TL;DR

This paper investigates the etale index of Brauer classes on Serre-Godeaux varieties, revealing that it can differ from the period, and applies this to compute indices in function fields using projective representation theory.

Contribution

It introduces methods combining Serre-Godeaux varieties, twisted K-theory, and representation theory to compute the etale index, showing it can differ from the period in specific cases.

Findings

The etale index can differ from the period of Brauer classes.

Explicit computations of the index in high-dimensional Serre-Godeaux varieties.

Application to unramified classes in function fields.

Abstract

We use the Serre-Godeaux varieties of finite groups, projective representation theory, the twisted Atiyah-Segal completion theorem, and our previous work on the topological period-index problem to compute the etale index of Brauer classes alpha in some specific examples. In particular, these computations show that the etale index of alpha differs from the period of alpha in general. As an application, we compute the index of unramified classes in the function fields of high-dimensional Serre-Godeaux varieties in terms of projective representation theory.