The Witt group of real algebraic varieties

The purpose of this paper is to compare the algebraic Witt group W(V ) of quadratic forms for an algebraic variety V over ℝ with a new topological invariant, WR(VCℂ), 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(Vℝ) of ℝ-linear topological vector bundles on the space Vℝ of real points of V . We show that the comparison maps W(V ) → WR(Vℂ) and WR(Vℂ) → KO(Vℝ) are isomorphisms modulo bounded 2-primary torsion.We give precise bounds for the exponent of the kernel and cokernel, depending upon the dimension of V. These results improve theorems of Knebusch, Mahé and Brumfiel. Along the way, we prove the 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.