On Generalized Sethi-vafa-witten Formulas

We present a formula for computing proper pushforwards of classes in the Chow ring of a projective bundle under the projection π : P(E )→ 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.