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

TL;DR

This paper compares two types of $L$-classes for projective varieties, showing when they coincide or differ, with results expressed in terms of Hodge signatures and singularity properties.

Contribution

It establishes conditions under which the intersection complex $L$-classes and constant coefficient $L$-classes differ, linking this to Hodge signatures of singularities.

Findings

$L_*(X)$ and $L^c_*(X)$ coincide for $Q$-homology manifolds.

Differences are characterized by Hodge signatures at singular points.

For hypersurfaces with isolated singularities, differences relate to link cohomology signatures.

Abstract

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.