The Witt group of real algebraic varieties

TL;DR

This paper compares algebraic and topological Witt groups of real algebraic varieties, establishing isomorphisms modulo 2-primary torsion and providing bounds on their differences, thus extending previous theorems in the field.

Contribution

It introduces a new topological invariant $WR(V_{\mathbb C})$ and proves its relation to algebraic Witt groups and $KO$-theory, improving known results and generalizing existing theorems.

Findings

The comparison maps are isomorphisms modulo bounded 2-primary torsion.

Provides explicit bounds for the kernel and cokernel exponents depending on the variety's dimension.

Establishes a comparison theorem between algebraic and topological Hermitian K-theory.

Abstract

Let $V$ be an algebraic variety over $\mathbb R$. The purpose of this paper is to compare its algebraic Witt group $W(V)$ with a new topological invariant $WR(V_{\mathbb C})$, based on symmetric forms on Real vector bundles (in the sense of Atiyah) on the space of complex points of $V$, This invariant lies between $W(V)$ and the group $KO(V_{\mathbb R})$ of $\mathbb R$-linear topological vector bundles on $V_{\mathbb R}$, the set of real points of $V$. We show that the comparison maps $W(V)\to WR(V_{\mathbb C})$ and $WR(V_{\mathbb C})\to KO(V_{\mathbb R})$ that we define are isomorphisms modulo bounded 2-primary torsion. We give precise bounds for the exponent of the kernel and cokernel of these maps, depending upon the dimension of $V.$ These results improve theorems of Knebusch, Brumfiel and Mah\'e. Along the way, we prove a comparison theorem between algebraic and topological Hermitian $K$-theory, and homotopy fixed point theorems for the latter. We also give a new proof (and a generalization) of a theorem of Brumfiel.