Hodge modules and cobordism classes
| Autor: | de Bobadilla, Javier Fernández, Pallarés, Irma, Saito, Morihiko |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We show that the cobordism class of a polarization of Hodge module defines a natural transformation from the Grothendieck group of Hodge modules to the cobordism group of self-dual bounded complexes with real coefficients and constructible cohomology sheaves in a compatible way with pushforward by proper morphisms. This implies a new proof of the well-definedness of the natural transformation from the Grothendieck group of varieties over a given variety to the above cobordism group (with real coefficients). As a corollary, we get a slight extension of a conjecture of Brasselet, Sch\"urmann and Yokura, showing that in the $\mathbb Q$-homologically isolated singularity case, the homology $L$-class which is the specialization of the Hirzebruch class coincides with the intersection complex $L$-class defined by Goresky, MacPherson, and others if and only if the sum of the reduced modified Euler-Hodge signatures of the stalks of the shifted intersection complex vanishes. Here Hodge signature uses a polarization of Hodge structure, and it does not seem easy to define it by a purely topological method. Comment: 21 pages |
| Databáze: | arXiv |
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