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

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 -equivariant cohomology [19]. Using the finite analytic correspondences from [7] we provide a sheaf-theoretic approach to ordinary equivariant -graded cohomology for any finite group . In particular, this gives a complex of sheaves 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}XX(\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