Irma Pallarés | Searcharxiv

arXiv2026

Constant coefficient and intersection complex $L$-classes of projective varieties

For a projective variety $X$, we have the intersection complex $L$-classes $L_*(X)$ defined by Goresky-MacPerson using cohomotopy and also the constant coefficient $L$-class $L^c_*(X)$ defined by applying an $L$-class transformation (or $T_{1*}$) to a cubic hyperresolution of $X$. These coincide if $X$ is a $\mathbb Q$-homology manifold. We show that the two $L$-classes $L_*(X)$ and $L^c_*(X)$ differ if they do by replacing $X$ with an intersection of general hyperplane sections which has only $\mathbb Q$-homologically isolated singularities. Finding a good sufficient condition for the non-coincidence of $L_*(X)$ and $L^c_*(X)$ is thus reduced to the latter case, where a necessary and sufficient condition has been obtained in terms of the Hodge signatures of stalks of intersection complex in our previous paper. In the case of projective hypersurfaces having only isolated singularities, the difference between $L_*(X)$ and $L^c_*(X)$ is given by the Hodge signatures of the link cohomologies at singular points, and the Hodge signatures of the vanishing cohomologies give the difference between $L^c_*(X)$ and the virtual $L$-class of $X$, that is, the image by a retraction map of the $L$-class of a smooth deformation of $X$ in an ambient smooth projective variety $Y$ in the very ample case.

math.AG↗

arXiv2022

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, Schürmann 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.

math.AG↗