1997/01/01 by Burt Totaro · 4 citations
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number #Cobordism #Commutative Algebra and Its Applications #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Pure mathematics #Torsion (gastropod)
paper · pdf · doi:10.1090/s0894-0347-97-00232-4
openalex publication_date 1997/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15
Atiyah and Hirzebruch gave the first counterexamples to the Hodge conjecture with integer coefficients. In particular, there is a smooth complex projective variety X of dimension 7 and a torsion element of H4 (X,Z) which is not the class of a codimension-2 algebraic cycle [4]. In this paper, we provide a more systematic explanation for their examples: for every smooth complex algebraic variety X, we show that the cycle map, from the ring of cycles modulo algebraic equivalence on X to the integer cohomology of X, lifts canonically to a more refined topological invariant of X, the ring MU ∗X ⊗MU ∗ Z, where MU∗X is the complex cobordism ring of X. Here MU ∗X is a module over the graded ring MU ∗ = MU ∗ (point) = Z[x1, x2,...], xi ∈ MU −2i, and we map MU ∗ to Z by sending all the generators xi to 0. The ring MU ∗X ⊗MU ∗ Z is the same as the integer cohomology ring if the integer cohomology is torsion-free, but in general the map MU ∗X ⊗MU ∗ Z → H ∗ (X,Z) need not be either injective or surjective, although the kernel and cokernel are torsion. This more refined cycle map gives a new way to prove that the Griffiths group (the kernel of the map from cycles modulo algebraic equivalence to integer cohomology) can be nonzero, without any use of Hodge theory. The resulting examples answer some questions on algebraic cycles by Colliot-Thélène and Schoen. Our examples are all quotients of complete intersections by finite groups, as are Atiyah-Hirzebruch’s examples. First, we find smooth complex projective varieties X of dimension 7, definable over Q, such that