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Let $E$ be a fiber $F$ bundle over a base $B$ such that $E, F$ and $B$ aresmooth compact complex algebraic varieties. In this paper we give explicitformulae for the difference of the Hirzebruch $\chi_y$-genus $\chi_y(E) -\chi_y(F)\chi_y(B)$. As a byproduct of the formulae we obtain that thesignature of such a fiber bundle is multiplicative mod $4$, i.e. the signaturedifference $\sigma(E) -\sigma(F)\sigma(B)$ is always divisible by $4$. In thecase of $\operatorname{dim}_{\mathbb C}E \leqq 4$ the $\chi_y$-genus difference$\chi_y(E) - \chi_y(F)\chi_y(B)$ can be concretely described only in terms ofthe signature difference $\sigma(E) -\sigma(F) \sigma(B)$ and/or the Todd genusdifference $\tau(E) - \tau(F)\tau(B)$. Using this we can obtain that in orderfor $\chi_y$ to be multiplicative $\chi_y(E) = \chi_y(F)\cdot \chi_y(B)$ forany such fiber bundle $y$ has to be $-1$, namely only the Euler-Poincar\'echaracteristic is multiplicative for any such fiber bundle.

Hirzebruch $\chi_y$-genera of complex algebraic fiber bundles -- the multiplicativity of the signature modulo $4$ --