The period-index problem for real surfaces

TL;DR

This paper investigates the relationship between period and index of Brauer group classes on real algebraic surfaces, establishing conditions for their equality and applying this to determine the u-invariant of the function field.

Contribution

It introduces a new Hodge-theoretic approach to the period-index problem for real surfaces and provides a complete characterization of when the period and index coincide.

Findings

Period equals index if the surface has no real points.

Necessary and sufficient condition for unramified classes.

The u-invariant of the function field is 4.

Abstract

We study when the period and the index of a class in the Brauer group of the function field of a real algebraic surface coincide. We prove that it is always the case if the surface has no real points (more generally, if the class vanishes in restriction to the real points of the locus where it is well-defined), and give a necessary and sufficient condition for unramified classes. As an application, we show that the u-invariant of the function field of a real algebraic surface is equal to 4, answering questions of Lang and Pfister. Our strategy relies on a new Hodge-theoretic approach to de Jong's period-index theorem on complex surfaces.