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

Abstract

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