Torsion algebraic cycles and complex cobordism. (English) Zbl 0989.14001

Totaro, Burt

J. Am. Math. Soc. 10, No. 2, 467-493 (1997).

From the introduction: M. Atiyah and F. Hirzebruch gave the first counterexamples to the Hodge conjecture with integral coefficients [Topology 1, 25-45 (1962; Zbl 0108.36401)]. 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.
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 non-injective, as well as non-surjective. The kernel of the classical cycle map is called the Griffiths group, and the topological proof here that the Griffiths group can be non-zero is the first proof of this fact which does not use Hodge theory. (The proof here gives non-zero torsion elements in the Griffiths group, whereas Griffiths’s Hodge-theoretic proof gives non-torsion elements.)
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-Thélène and Schoen [see J.-L. Colliot-Thélène in: Arithmetic algebraic geometry, Trento 1991, Lect. Notes Math. 1553, 1-49 (1993; Zbl 0806.14002), p. 14, and C. Schoen, Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, No. 1, 1-50 (1995; Zbl 0839.14004), p. 13]. Colliot-Thélène asked, in particular, whether the map CH\(^2(X)/n \to H^4(X,\mathbb{Z}/n)\) is injective for all smooth complex projective varieties \(X\). Here \(\text{CH}^iX\) is the group of codimension \(i\) algebraic cycles modulo rational equivalence. The first examples where Colliot-Thélène’s map is not injective were found by Kollár and van Geemen, very recently, Bloch and Esnault found examples defined over number fields. Here our topological method gives examples which can be defined over the rational numbers. The varieties we use, as in Atiyah-Hirzebruch’s examples, are quotients of complete intersections by finite groups. C. Schoen (loc. cit.) asked whether the map from the torsion subgroup of \(\text{CH}^iX\) to Deligne cohomology is injective for all smooth complex projective varieties \(X\). This was known in many cases: for \(i\leq 2\) by Merkur’ev-Suslin, and for \(i=\dim X\) by Roitman. But we show that injectivity can fail for \(i=3\). Similarly, one can ask whether an algebraic cycle which maps to 0 in Deligne cohomology must be algebraically equivalent to 0, the point being that Griffiths’s original examples of cycles which were homologically but not algebraically equivalent to 0 had non-zero image in Deligne cohomology. The answer is no, as M. Nori showed by a subtle application of Hodge theory [Invent. Math. 111, No. 2, 349-373 (1993; Zbl 0822.14008)]. Here we show again that the answer is no, using our topological method. It is interesting that both Nori’s examples and ours work in codimension at least 3, Nori suggests that the answer to the question should be yes for codimension 2 cycles.

MSC:

14C25	Algebraic cycles
55N22	Bordism and cobordism theories and formal group laws in algebraic topology

Keywords:

algebraic cycles; cycle map; Griffiths group; complex cobordism; integral cohomology

Citations:

Zbl 0108.36401; Zbl 0806.14002; Zbl 0839.14004; Zbl 0822.14008

References:

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