JournalscmhVol. 92, No. 3pp. 429–465

An explicit cycle map for the motivic cohomology of real varieties

  • Pedro F. dos Santos

    Instituto Superior Técnico, Lisbon, Portugal
  • Robert M. Hardt

    Rice University, Houston, USA
  • James D. Lewis

    University of Alberta, Edmonton, Canada
  • Paulo Lima-Filho

    Texas A&M University, College Station, USA
An explicit cycle map for the motivic cohomology of real varieties cover

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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 Gal(C/R)\operatorname{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)RO(G)-graded cohomology for any finite group GG. In particular, this gives a complex of sheaves Zpω\mathbb Zp_{\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. Our cycle map is induced by the change of sites functor that assigns to a real variety Xitsanalyticspaceits analytic spaceX(\mathbb C)$ together with the complex conjugation involution.

Cite this article

Pedro F. dos Santos, Robert M. Hardt, James D. Lewis, Paulo Lima-Filho, An explicit cycle map for the motivic cohomology of real varieties. Comment. Math. Helv. 92 (2017), no. 3, pp. 429–465

DOI 10.4171/CMH/416