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I. Hambleton, A. Korzeniewski and A. Ranicki proved that the signature of afibre bundle of closed, connected, compatibly oriented PL manifolds is alwaysmultiplicative modulo 4. In this paper, we consider the Hirzebruch$\chi_y$-genera for odd integers $y$ for a smooth fiber bundle such that thebase, fibre, and total space are compact complex algebraic manifolds (in thecomplex analytic topology, not in the Zariski topology). We show that theHirzebruch $\chi_y$-genera of such a fibre bundle are always multiplicativemodulo 4. We also investigate multiplicativity modulo 8 and show that if $y$ iscongruent to 3 modulo 4, then the $\chi_y$-genera are multiplicative modulo 8.We also show that when $y$ is congruent to 1 modulo 4, the Hirzebruch$\chi_y$-genera of such a fiber bundle are multiplicative modulo 8 if and onlyif the signature is multiplicative modulo 8, and that the non-multiplicativitymodulo 8, in this case, is identified with an Arf-Kervaire invariant.

Hirzebruch $\chi_y$-genera modulo $8$ of fiber bundles for odd integers $y$