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 ( V 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 K O ( V R ) of R-linear topological vector bundles on the space V R of real points of V. We show that the comparison maps W ( V ) → WR ( V C ) and WR ( V C ) → K O ( V R ) 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.