On Hirzebruch invariants of elliptic fibrations

TL;DR

This paper calculates all Hirzebruch invariants for various elliptic fibrations across all dimensions, providing generating series that relate these invariants to base invariants.

Contribution

It introduces a unified generating series for Hirzebruch invariants of D5, E6, E7, and E8 elliptic fibrations in any dimension, linking them to base invariants.

Findings

Generated series encode all Hirzebruch invariants for the fibrations.

Explicit formulas relate invariants to base properties.

Applicable to fibrations of arbitrary dimension.

Abstract

We compute all Hirzebruch invariants $\chi_q$ for $D_5$, $E_6$, $E_7$ and $E_8$ elliptic fibrations of every dimension. A single generating series $\chi(t,y)$ is produced for each family of fibrations such that the coefficient of $t^{k}y^{q}$ encodes $\chi_q$ over a base of dimension $k$, solely in terms of invariants of the base of the fibration.