Hodge modules and cobordism classes

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.