For a pair (f, g) of morphisms f:X→Z\documentclass[12pt]{minimal}
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\begin{document}$$f:X \rightarrow Z$$\end{document} and g:Y→Z\documentclass[12pt]{minimal}
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\begin{document}$$g:Y \rightarrow Z$$\end{document} of (possibly singular) complex algebraic varieties X, Y, Z, we present congruence formulae for the difference f∗Ty∗(X)-g∗Ty∗(Y)\documentclass[12pt]{minimal}
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\begin{document}$$f_*T_{y*}(X) -g_*T_{y*}(Y)$$\end{document} of pushforwards of the corresponding motivic Hirzebruch classes Ty∗\documentclass[12pt]{minimal}
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\begin{document}$$T_{y*}$$\end{document}. If we consider the special pair of a fiber bundle F↪E→B\documentclass[12pt]{minimal}
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\begin{document}$$F \hookrightarrow E \rightarrow B$$\end{document} and the projection pr2:F×B→B\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{pr}_2:F {\times }B \rightarrow B$$\end{document} as such a pair (f, g), then we get a congruence formula for the difference f∗Ty∗(E)-χy(F)Ty∗(B)\documentclass[12pt]{minimal}
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\begin{document}$$f_*T_{y*}(E) -\chi _y(F)T_{y*}(B)$$\end{document}, which at degree level yields a congruence formula for χy(E)-χy(F)χy(B)\documentclass[12pt]{minimal}
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\begin{document}$$\chi _y(E) -\chi _y(F)\chi _y(B)$$\end{document}, expressed in terms of the Euler–Poincaré 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.
