Higher discriminants and the topology of algebraic maps
Higher discriminants and the topology of algebraic maps
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更高的判别式和代数图的拓扑
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
V. Shende
中科院分区：
文献类型：
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作者：
L. Migliorini;V. Shende
我们证明了复代数变种的固有族中Betti上同调的变化方式是由基中的某些“高判判式”控制的。这些判别式是根据横向条件定义的，在光滑变体之间的态射的情况下，可以通过切空间计算来检查。它们在以下两种意义上控制上同调的变化：(1)IC轴沿投影映射的任意推前和的支撑点是一个高判别式的分量；(2)常数函数的适当推前的特征循环的任何分量是一个高判别式分量的正规变化。
We show that the way in which Betti cohomology varies in a proper family of complex algebraic varieties is controlled by certain "higher discriminants" in the base. These discriminants are defined in terms of transversality conditions, which in the case of a morphism between smooth varieties can be checked by a tangent space calculation. They control the variation of cohomology in the following two senses: (1) the support of any summand of the pushforward of the IC sheaf along a projective map is a component of a higher discriminant, and (2) any component of the characteristic cycle of the proper pushforward of the constant function is a conormal variety to a component of a higher discriminant.
The same would hold for the Whitney stratification of the family, but there are vastly fewer higher discriminants than Whitney strata. For example, in the case of the Hitchin fibration, the stratification by higher discriminants gives exactly the {delta} stratification introduced by Ngo.