The push-forward and Todd class of flag bundles

Abstract

Consider a connected reductive algebraic group G over an algebraically closed field k, and a principal G-bundle π : X → Y , where X and Y are non-singular algebraic varieties over k. For any parabolic subgroup P ⊂ G, the map π factors through the flag bundle h : X/P → Y . In this note, we describe the push-forward (or Gysin homomorphism) h∗ : A∗(X/P ) → A∗(Y ) where A∗ denotes the Chow group. Moreover, we compute the Todd class of the tangent bundle to h in A∗(X/P )Q. In the case when k is the field of complex numbers, our results hold when the Chow ring is replaced by the rational cohomology ring, and the proofs are the same. The push-forward is described in [P] when G is the general linear group, and in [AC] for the canonical map G/B → G/P where G is arbitrary and B is a Borel subgroup of P . Note that this map is a flag bundle associated with the principal P/R(P )-bundle G/R(P ) → G/P , where R(P ) denotes the radical of P . Our formula for the Todd class seems to be new.