The Witt group of real algebraic varieties

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 Mahé. 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.

References in corpus (3)

• Hermitian K-theory, derived equivalences and Karoubi's Fundamental Theorem
• Person Re-identification Meets Image Search
• Periodicity of hermitian K-theory and Milnor's K-groups

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