Witt groups of smooth real curves and surfaces

Abstract

We study the $\mathbf{I}^*$-cohomology of a smooth real algebraic curve in terms of its real locus and its geometric genus. We notably extend results of Monnier to the twisted case, which is crucial to the understanding of proper pushforwards of Witt groups. We also perform some computations related to transfers along the finite \'etale extension $\mathbb{C}/\mathbb{R}$. We further describe how to compute twisted Witt groups of surfaces, extending work of Sujatha, and the image of the global signature homomorphism following Monnier. As an application of the main methods of the paper, we describe the shifted and twisted Witt groups of smooth anisotropic quadrics over $\mathbb{R}$ of dimension $\leq 3$.