Homological congruence formulae for characteristic classes of singular varieties

TL;DR

This paper develops homological congruence formulas for characteristic classes of singular varieties, generalizing classical invariants and extending results to singular complex projective varieties using intersection cohomology.

Contribution

It introduces new congruence formulas for motivic Hirzebruch classes of singular varieties, extending classical identities to the singular setting with intersection cohomology techniques.

Findings

Derived congruence formulas for motivic Hirzebruch classes of singular varieties.

Extended Rovi--Yokura identities to singular complex projective varieties.

Expressed differences of characteristic classes in terms of classical invariants.

Abstract

For a pair $(f, g)$ of morphisms $f:X \to Z$ and $g:Y \to Z$ of (possibly singular) complex algebraic varieties $X,Y,Z$, we present congruence formulae for the difference $f_*T_{y*}(X) -g_*T_{y*}(Y)$ of pushforwards of the corresponding motivic Hirzebruch classes $T_{y*}$. If we consider the special pair of a fiber bundle $F \hookrightarrow E \to B$ and the projection $pr_2:F \times B \to B$ as such a pair $(f,g)$, then we get a congruence formula for the difference $f_*T_{y*}(E) -\chi_y(F)T_{y*}(B)$, which at degree level yields a congruence formula for $\chi_y(E) -\chi_y(F)\chi_y(B)$, expressed in terms of the Euler--Poincarv'e characteristic, Todd genus and signature in the case when $F, E, B$ are non-singular and compact. We also extend the finer congruence identities of Rovi--Yokura to the singular complex projective situation, by using the corresponding intersection (co)homology invariants.