Torsion algebraic cycles and complex cobordism

Abstract

Atiyah and Hirzebruch gave the first counterexamples to the Hodge conjecture with integral coefficients [3]. That conjecture predicted that every integral cohomology class of Hodge type (p, p) on a smooth projective variety should be the class of an algebraic cycle, but Atiyah and Hirzebruch found additional topological properties which must be satisfied by the integral cohomology class of an algebraic cycle. Here we provide a more systematic explanation for their results by showing that the classical cycle map, from algebraic cycles modulo algebraic equivalence to integral cohomology, factors naturally through a topologically defined ring which is richer than integral cohomology. The new ring is based on complex cobordism, a well-developed topological theory which has been used only rarely in algebraic geometry since Hirzebruch used it to prove the Riemann-Roch theorem [17]. This factorization of the classical cycle map implies the topological restrictions on algebraic cycles found by Atiyah and Hirzebruch. It goes beyond their work by giving a topological method to show that the classical cycle map can be noninjective, as well as nonsurjective. The kernel of the classical cycle map is called the Griffiths group, and the topological proof here that the Griffiths group can be nonzero is the first proof of this fact which does not use Hodge theory. (The proof here gives nonzero torsion elements in the Griffiths group, whereas Griffiths's Hodge-theoretic proof gives nontorsion elements [13].) This topological argument also gives examples of algebraic cycles in the kernel of various related cycle maps where few or no examples were known before, thus answering some questions posed by Colliot-Thelene and Schoen ([8], p. 14; [37], p. 13). Colliot-Thelene asked, in particular, whether the map CH2(X)/n -* H4(X, Z/n) is injective for all smooth complex projective varieties X. Here CH'X is the group of codimension i algebraic cycles modulo rational equivalence. The first examples where Colliot-Thelene's map is not injective were found by Kollar and van Geemen (see [4], p. 135); very recently, Bloch and Esnault found examples defined over number fields [7]. (Over nonalgebraically closed fields k there are other examples of smooth projective varieties Xk with CH2 (Xk)/n -* Hg4t(Xk, Z/n) not injective, due to Colliot-Thelene and Sansuc as reinterpreted by Salberger (see [9] and [8], Remark 7.6.1), and Parimala and Suresh [31]. These elements of CH2 (Xk)/n are not shown to remain nonzero in CH2(Xc)/n, however.) Here our topological method