Torsion algebraic cycles and complex cobordism
Torsion algebraic cycles and complex cobordism
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扭转代数环和复配边
DOI：
10.1090/s0894-0347-97-00232-4
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发表时间：
1996
影响因子：
3.9
通讯作者：
B. Totaro
中科院分区：
文献类型：
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作者：
B. Totaro
Atiyah 和 Hirzebruch 用积分系数给出了霍奇猜想的第一个反例 [3]。该猜想预测，光滑射影簇上的霍奇型 (p, p) 的每个积分上同调类都应该是代数圈的类，但 Atiyah 和 Hirzebruch 发现了代数圈的积分上同调类必须满足的附加拓扑性质。在这里，我们为他们的结果提供了更系统的解释，通过显示经典循环图，从代数循环模代数等价到积分上同调，自然地通过比积分上同调更丰富的拓扑定义的环进行因子分解。新环基于复配边，这是一种成熟的拓扑理论，自从 Hirzebruch 用它来证明黎曼-罗赫定理以来，该理论很少在代数几何中使用[17]。经典循环图的这种因式分解意味着 Atiyah 和 Hirzebruch 发现的代数循环的拓扑限制。它超越了他们的工作，提供了一种拓扑方法来表明经典循环图可以是非内射的，也可以是非满射的。经典循环图的核心称为Griffiths群，这里Griffiths群可以非零的拓扑证明是该事实的第一个不使用Hodge理论的证明。 （这里的证明给出了格里菲斯群中的非零扭转元素，而格里菲斯的霍奇理论证明给出了非扭转元素[13]。）这个拓扑论证还给出了各种相关循环图的核心中的代数循环的例子，其中以前很少或根本没有例子，从而回答了Colliot-Thelene和Schoen提出的一些问题（[8]，第14页；[37]，第13页）。 Colliot-Thelene 特别询问映射 CH2(X)/n -* H4(X, Z/n) 对于所有光滑复射影簇 X 是否是内射的。这里 CH'X 是代数环模有理等价的余维群。 Colliot-Thelene 映射不是单射的第一个例子是由 Kollar 和 van Geemen 发现的（参见 [4]，第 135 页）；最近，Bloch 和 Esnault 发现了在数字字段上定义的示例 [7]。 （在非代数闭域 k 上，还有其他平滑射影簇 Xk 的例子，其中 CH2 (Xk)/n -* Hg4t(Xk, Z/n) 不是单射的，这是由于 Salberger 重新解释的 Colliot-Thelene 和 Sansuc（参见 [9] 和 [8]，备注 7.6.1）以及 Parimala 和 Suresh [31]。CH2 (Xk)/n 的这些元素并未显示为保留然而，CH2(Xc)/n 中非零。）这是我们的拓扑方法
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