An explicit cycle map for the motivic cohomology of real varieties

Abstract

We provide a direct construction of a cycle map in the level of representing complexes from the motivic cohomology of real (or complex) varieties to the appropriate ordinary cohomology theory. For complex varieties, this is simply integral Betti cohomology, whereas for real varieties the recipient theory is the bigraded $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$-equivariant cohomology [19]. Using the finite analytic correspondences from [7] we provide a sheaf-theoretic approach to ordinary equivariant $RO(G)$-graded cohomology for any finite group $G$. In particular, this gives a complex of sheaves $\mathbb{Z}{p}_{\omega }$ on a suitable equivariant site of real analytic manifolds-with-corner whose construction closely parallels that of the Voevodsky's motivic complexes $$\mathbb Zp_{\mathcal M}$.Ourcyclemapisinducedbythechangeofsitesfunctorthatassignstoarealvariety$X$itsanalyticspace$X(\mathbb C)$ together with the complex conjugation involution.