- journal article metadata
European Mathematical Society Publishing House
2017-07-30 23:45:00
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
92
2017
3
An explicit cycle map for the motivic cohomology of real varieties
Pedro
dos Santos
Instituto Superior Técnico, Lisbon, Portugal
Robert
Hardt
Rice University, Houston, USA
James
Lewis
University of Alberta, Edmonton, Canada
Paulo
Lima-Filho
Texas A&M University, College Station, USA
Ordinary equivariant cohomology, motivic cohomology, cycle map, finite analytic currents, real varieties
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 $\operatorname{Gal}(\mathbb C/\mathbb R)$-equivariant cohomology [19]. Using the finite analytic correspondences from [7] we provide a sheaf-theoretic approach to ordinary equivariant $RO(G)$-graded cohomology for any finite group $G$. In particular, this gives a complex of sheaves $\mathbb Zp_{\omega}$ 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}$. Our cycle map is induced by the change of sites functor that assigns to a real variety $X$ its analytic space $X(\mathbb C)$ together with the complex conjugation involution.
Algebraic geometry
Several complex variables and analytic spaces
Calculus of variations and optimal control; optimization
Algebraic topology
429
465
10.4171/CMH/416
http://www.ems-ph.org/doi/10.4171/CMH/416