Lawson同调上的一个次级乘法结构
A Secondary Multiplicative Structure on Lawson Homology
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中文总结 AI 辅助
本文在光滑复射影簇的Lawson同调上引入一种次级乘法运算，细化交积并产生非平凡的整系数Massey括号，其值在奇异同调中消失，且在射影丛等映射下保持，给出形式性障碍。
中文摘要 AI 辅助
我们引入了一个关于光滑复射影簇的Lawson同调的新次级运算。它通过保留在分次Lawson交积代数层面上不可见的零同伦数据，细化了Lawson交积。本质上，该运算是乘法态射上同调中的Toda括号，经由Friedlander–Lawson对偶性传递而来；在结合余链模型中，它由经典的Massey公式表示。对于每个定义的三元组，仿射括号模去其全部不确定性给出一个良定义的次级不变量。这一新结构是真正非平凡的，并且严格细于其奇异同调实现。在一个光滑射影八维簇上，我们构造了整系数Lawson–Massey括号，其商类非零，而特定值具有精确的阶二，并在Lawson环映射到奇异同调下映射为零。一个平方零扰动使得所有三个相关的奇异同调类变为非零，而不改变次级值。即使在一个固定的奇异同调三元组上，对所有这些不确定性的和取商之后，仍有无限多个这样的值保持独立。投影公式传递在射影丛和奇数次一般有限态射下保持该现象，而循环三重覆盖在至少八维的每个维度上给出了具有丰富典范丛的例子。相同的定义系统产生了非零的动机Massey值、标量不可分解的高次Chow挠元，以及对整形式和二局部形式性的障碍。
英文摘要
We introduce a new secondary operation on Lawson homology of smooth complex projective varieties. It refines the Lawson intersection product by retaining null-homotopy data that are invisible at the level of the graded Lawson intersection algebra. Intrinsically, the operation is the Toda bracket in multiplicative morphic cohomology transported through Friedlander--Lawson duality; in an associative cochain model it is represented by the classical Massey formula. For every defined triple, the affine bracket modulo its full indeterminacy gives a well-defined secondary invariant. The new structure is genuinely nontrivial and strictly finer than its singular-homology realization. On a smooth projective eightfold we construct integral Lawson--Massey brackets whose quotient classes are nonzero, while specified values have exact order two and map to zero under the Lawson cycle map to singular homology. A square-zero perturbation makes all three associated singular-homology classes nonzero without changing the secondary values. Infinitely many such values remain independent even after quotienting by the sum of all their indeterminacies, over one fixed singular-homology triple. Projection-formula transfers preserve the phenomenon under projective bundles and odd-degree generically finite morphisms, and cyclic triple covers give examples with ample canonical bundle in every dimension at least eight. The same defining systems yield nonzero motivic Massey values, scalar-indecomposable higher Chow torsion, and obstructions to integral and two-local formality.
发表机构
- Sichuan University（四川大学）
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