Witt groups of smooth real curves and surfaces

Abstract

We study the ${\mathbf{I}}^{\ast }$-cohomology of a smooth real algebraic curve in terms of its real locus and its geometric genus. We notably extend results of Knebusch and Monnier to the twisted case. We further describe how to compute twisted Witt groups of surfaces, generalising work of Sujatha, and apply our results to the computation of the shifted and twisted Witt groups of the smooth real anisotropic quadric surface. Finally, we extend Monnier's investigation of the image of the global signature homomorphism for surfaces.

arXiv comment31 pages. v4: greatly streamlined, study of (Chow--)Witt groups of quadrics reduced to the Witt groups of the smooth real anisotropic surface, removed investigation of Pardon spectral sequence, comments still welcome