Characteristic classes and quadric bundles

TLDR

On complex varieties, these classes correspond to the pullbacks of the usual Stiefel‑Whitney and Euler classes from the cohomology of the classifying space BSO(N,ℂ). The paper constructs Stiefel‑Whitney and Euler classes in Chow cohomology for algebraic vector bundles equipped with a nondegenerate quadratic form. Using the geometry of quadric bundles, the authors study Chern classes of maximal isotropic subbundles, define the new classes as Chow cohomology classes pulling back to those Chern classes, and obtain Schubert‑type presentations. These classes are new characteristic classes in algebraic geometry, not generated by Chern classes, satisfy ci(E)=ci(F) mod 2 and cn(E)=±cn(F), confirm Fulton’s conjecture, and exist uniquely even when a maximal isotropic subbundle is absent.

Abstract

In this paper we construct Stiefel-Whitney and Euler classes in Chow cohomology for algebraic vector bundles with nondegenerate quadratic form. These classes are not in the algebra generated by the Chern classes of such bundles and are new characteristic classes in algebraic geometry. On complex varieties, they correspond to classes with the same name pulled back from the cohomology of the classifying space BSO(N,C). The classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups ([T2], [EG]). We begin by using the geometry of quadric bundles to study Chern classes of maximal isotropic subbundles. If V → X is a vector bundle with quadratic form, and if E and F are maximal isotropic subbundles of V then we prove (Theorem 1) that ci(E) and ci(F ) are equal mod 2. Moreover, if the rank of V is 2n, then cn(E) = ±cn(F ), proving a conjecture of Fulton. We define Stiefel-Whitney and Euler classes as Chow cohomology classes which pull back to Chern classes of maximal isotropic subbundles of the pullback bundle. Using the above theorem we show (Theorem 2) that these classes exist and are unique, even though V need not have a maximal isotropic subbundle. These constructions also make it possible to give “Schubert” presentations,