Homological congruence formulae for characteristic classes of singular varieties

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 $f_{\ast }{T}_{y\ast }(X)-g_{\ast }{T}_{y\ast }(Y)$ of pushforwards of the corresponding motivic Hirzebruch classes $T_{y\ast }$. If we consider the special pair of a fiber bundle $F\hookrightarrow E\to B$ and the projection $pr_2:F \times B \to B$assuchapair$(f,g)$, then we get a congruence formula for the difference $f_{\ast }{T}_{y\ast }(E)-{\chi }_y(F)T_{y\ast }(B)$, 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.