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

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.