The Wu relations in real algebraic geometry

TL;DR

This paper explores the relationships between Chern classes and Galois cohomology in real algebraic varieties without real points, with applications to topology and sums of squares problems.

Contribution

It introduces new relations in equivariant cohomology for such varieties and applies these to solve problems in sums of squares and topology.

Findings

-1 is a sum of 2 squares in the function field of certain real algebraic surfaces

Extended results to higher-dimensional varieties with similar properties

Provides new tools linking algebraic geometry and real algebraic topology

Abstract

We construct and study relations between Chern classes and Galois cohomology classes in the Gal(C/R)-equivariant cohomology of real algebraic varieties with no real points. We give applications to the topology of their sets of complex points, and to sums of squares problems. In particular, we show that -1 is a sum of 2 squares in the function field of any smooth projective real algebraic surface with no real points and with vanishing geometric genus, as well as higher-dimensional generalizations of this result.