Hodge modules and cobordism classes

  • Javier Fernández de Bobadilla

    Basque Center for Applied Mathematics,48009 Bilbao, Spain
  • Irma Pallarés

    Department of Mathematics, KU Leuven,3001 Leuven, Belgium
  • Morihiko Saito

    Department of Mathematics, Kyoto University, RIMS Kyoto University, Kyoto 606-8502, Japan
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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 push-forward 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, Schürmann and Yokura, showing that in the -homologically isolated singularity case, the homology -class which is the specialization of the Hirzebruch class coincides with the intersection complex -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.

Cite this article

Javier Fernández de Bobadilla, Irma Pallarés, Morihiko Saito, Hodge modules and cobordism classes. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1387