From the Global Signature to Higher Signatures

Let X be an algebraic variety over the field of real numbers R. We use the signature of a quadratic form to produce “higher” global signatures relating the derived Witt groups of X to the singular cohomology of the real points X(R) with integer coefficients. We also study the global signature ring homomorphism and use the powers of the fundamental ideal in the Witt ring to prove an integral version of a theorem of Raman Parimala and Jean ColliotThelene on the mod 2 signature. Furthermore, we obtain an Atiyah-Hirzebruch spectral sequence for the derived Witt groups of X with 2 inverted. Using this spectral sequence, we provide a bound on the ranks of the derived Witt groups of X in terms of the Betti numbers of X(R). We apply our results to answer a question of Max Karoubi on boundedness of torsion in the Witt group of X. Throughout the article, the results are proved for a wide class of schemes over an arbitrary base field of characteristic different from 2 using real cohomology in place of singular cohomology.