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

Abstract

Let $E$ be a fiber $F$ bundle over a base $B$ such that $E, F$ and $B$ are smooth compact complex algebraic varieties. In this paper we give explicit formulae for the difference of the Hirzebruch $χ_y$-genus $χ_y(E) - χ_y(F)χ_y(B)$. As a byproduct of the formulae we obtain that the signature of such a fiber bundle is multiplicative mod $4$, i.e. the signature difference $σ(E) -σ(F)σ(B)$ is always divisible by $4$. In the case of $\operatorname{dim}_{\mathbb C}E \leqq 4$ the $χ_y$-genus difference $χ_y(E) - χ_y(F)χ_y(B)$ can be concretely described only in terms of the signature difference $σ(E) -σ(F) σ(B)$ and/or the Todd genus difference $τ(E) - τ(F)τ(B)$. Using this we can obtain that in order for $χ_y$ to be multiplicative $χ_y(E) = χ_y(F)\cdot χ_y(B)$ for any such fiber bundle $y$ has to be $-1$, namely only the Euler-Poincaré characteristic is multiplicative for any such fiber bundle.