Characteristic classes in the Chow ring

TL;DR

This paper investigates the structure of characteristic classes in the Chow ring for principal G-bundles, revealing isomorphisms with Weyl group invariants and establishing algebraicity of certain topological classes.

Contribution

It demonstrates that for reductive groups, the characteristic class ring aligns with Weyl invariants, and extends this to general bundles over complex varieties after rationalization.

Findings

Ring of characteristic classes is isomorphic to Weyl invariants for reductive G.

The isomorphism extends to general bundles after tensoring with Q.

Topological characteristic classes are algebraic for Zariski locally trivial bundles over complex varieties.

Abstract

We study the ring of characteristic classes with values in the Chow ring for principal $G$-bundles over schemes. If we consider bundles which are locally trivial in the Zariski topology, then we show, for $G$ reductive, that this ring is isomorphic to the Weyl group invariants in the algebra generated by characters of the maximal torus. For general principal bundles the same isomorphism holds after tensoring the coefficients with ${\Bbb Q}$. As a corollary, we show that any (non-torsion) topological characteristic class is algebraic when applied to Zariski locally trivial bundles over complex algebraic varieties.