On generalized Sethi-Vafa-Witten formulas

TL;DR

This paper derives a general formula for pushforwards in the Chow ring of projective bundles, enabling new calculations of topological invariants of elliptically fibered Calabi-Yau fourfolds relevant to string theory compactifications.

Contribution

It introduces a unified formula for Chow ring pushforwards that generalizes previous Sethi-Vafa-Witten formulas for elliptic fibrations in algebraic geometry.

Findings

The formula computes Chern numbers of complete intersections in projective bundles.

It generalizes existing formulas for Euler characteristics of elliptic Calabi-Yau fourfolds.

Applications include calculating topological invariants relevant to F-theory.

Abstract

We present a formula for computing proper pushforwards of classes in the Chow ring of a projective bundle under the projection $\pi:\Pbb(\Escr)\rightarrow B$, for $B$ a non-singular compact complex algebraic variety of any dimension. Our formula readily produces generalizations of formulas derived by Sethi,Vafa, and Witten to compute the Euler characteristic of elliptically fibered Calabi-Yau fourfolds used for F-theory compactifications of string vacua. The utility of such a formula is illustrated through applications, such as the ability to compute the Chern numbers of any non-singular complete intersection in such a projective bundle in terms of the Chern class of a line bundle on $B$.