Torsion algebraic cycles and complex cobordism

1996/09/23 by Burt Totaro, Totaro, Burt
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #alg-geom #math.AG

paper · pdf · doi:10.48550/arxiv.alg-geom/9609016

20 pages

openalex publication_date 1996/09/23 · arxiv created 1996/09/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the cycle map on a variety X, from algebraic cycles modulo algebraic equivalence to integer cohomology, lifts canonically to a topologically defined quotient of the complex cobordism ring of X. This more refined cycle map gives a topological proof that the Griffiths group is nonzero for some varieties X, without any use of Hodge theory. We also use this more refined cycle map to give examples of torsion algebraic cycles which map to 0 in Deligne cohomology but are not algebraically equivalent to 0, thus answering some questions by Colliot-Thelene and Schoen.

Citations

• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I

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