Torsion algebraic cycles and complex cobordism

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 H(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 ofX , lifts canonically to a more refined topological invariant of X , the ring MUX⊗MU∗ Z, where MU X is the complex cobordism ring of X . Here MUX is a module over the graded ring MU = MU(point) = Z[x1, x2, . . . ], xi ∈ MU , 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 MUX⊗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 the map CHX/2 → H(X,Z/2) is not injective. Here CHX is the group of codimension i algebraic cycles on X modulo rational equivalence. Kollár and van Geemen [5], p. 135, gave the first examples of smooth complex projective varieties with CHX/n → H(X,Z/n) not injective for some n, answering a question by Colliot-Thélène [9], p. 14. Over non-algebraically closed fields k there are other examples of smooth projective varieties Xk with CH (Xk)/n → H 4 et(Xk,Z/n) not injective, due to Colliot-Thélène and Sansuc as reinterpreted by Salberger (see [10] and [9], Remark 7.6.1), Parimala and Suresh [29], and Bloch and Esnault [8]. Of these examples, only Bloch and Esnault’s elements of CH(Xk)/n are shown to remain nonzero in CH(Xk)/n, as happens in our example. Also, we find codimension-3 cycles, on certain smooth complex projective varieties X of dimension 15, which are torsion in the Chow group CHX , which map to 0 in H(X,Z) and even in Deligne cohomology (i.e., the intermediate Jacobian), but which are not algebraically equivalent to 0. The variety X and the cycles we consider can be defined over Q. By contrast, for all X over C, the map from the torsion subgroup of CHX to Deligne cohomology was known to be injective for i ≤ 2 by Merkur’ev-Suslin [22], p. 338, and for i = dim X by Roitman [34], and Schoen [35], p. 13, asked whether the map was injective in general. Similarly, it is conjectured that codimension-2 cycles which map to 0 in Deligne cohomology are algebraically equivalent to 0, and our construction shows that this is false in codimension 3, as Nori was the first to show [26]. (Nori’s cycles are non-torsion in the group of cycles modulo algebraic equivalence, in contrast to ours.) We now describe the argument in more detail. In general, if X is a complex algebraic scheme which can be singular or noncompact, the usual cycle class map sends the group Z i X of i-dimensional algebraic cycles modulo algebraic equivalence to the Borel-Moore homology H 2i (X,Z) [13], chapter 19, and we lift this map to the degree 2i subgroup of the graded group MU ∗ X ⊗MU∗ Z; we informally call that subgroup MU 2i X ⊗MU∗ Z. Here, for a locally compact space X , MU BM ∗ X denotes the Borel-Moore version of the complex bordism groups of X . That is, for X compact they are the usual complex bordism groups of X , generated by continuous maps of closed manifolds with a complex structure on their stable tangent bundle to X ; and if X = X −S for a CW complex X with a closed subcomplex S, then MU ∗ X = MU∗(X,S). The construction of the new cycle map uses Hironaka’s resolution of singularities together with some fundamental results on complex cobordism proved by Quillen and Wilson. IfX is a locally compact space such that H ∗ (X,Z) is torsion-free, then the natural mapMU BM ∗ X⊗MU∗ Z → H ∗ (X,Z) is an isomorphism; so the new cycle class map says nothing new for varietiesX with torsionfree homology. Also, the map MU ∗ X ⊗MU∗ Z → H BM ∗ (X,Z) is always an isomorphism after tensoring