Universal Gysin formulas for flag bundles

Abstract

Abstract. -We give push-forward formulas for all flag bundles of types A, B, C, D. The formulas (and also the proofs) involve only Segre classes of the original vector bundles and characteristic classes of universal bundles. As an application, we provide new determinantal formulas. IntroductionA proper morphism F : Y → X of non-singular algebraic varieties over an algebraically closed field yields an additive mapX of Chow groups induced by push-forward cycles, called the Gysin map (see Fulton's book [7]; note that the theory developed in this book allows one to generalize the results of the present paper to singular varieties over a field and their Chow groups; moreover, for complex varieties, one can also use the cohomology rings with integral coefficients). We will alternatively denote F * by X Y . Push-forward formulas show how the classes of algebraic cycles on Y go via the Gysin map to classes of algebraic cycles on X. In the present paper, we are interested in push-forwards in flag bundles. We shall give formulas for the classical types A, B, C, D. These have a universal character in three aspects: -these involve characteristic classes of universal vector bundles; -these universally hold for any polynomial in such classes; -these use in a universal way only the Segre classes of the original vector bundles.The starting point of our argument is a reformulation of the classical formula for push-forward of powers of the hyperplane class in a projective bundle, that we recall. Let E → X be a vector bundle of rank n on a variety X. Let P(E) → X be the projective bundle of lines in E and let ξ ≔ c 1 (O P(E) (1)) be the hyperplane class. For any i, the ith Segre class of E isThis is a definition in [7] and a lemma in preceding intersection theory (see e.g. [13, Lemma 1]). Then, consider a polynomial f (ξ) = i α i ξ i with coefficients α i in the Chow ring of X (here, we identify A• X with a subring of A • P(E) and throughout the text, we will often omit pullback notation for vector bundles and algebraic cycles). Using the projection formula

Gysin maps, duality, and Schubert classes

“…As in [DP15], one can deduce a Gysin formula for F µ (E • ) → X from the described structure of chain of projective bundles (5). For the sake of completeness, and also because we make different notational conventions, let us recall the main lines of the argument.…”

“…The present paper is a continuation of [DP15]. We apply Gysin formulas for flag bundles to study Schubert calculus on Grassmann bundles.…”

“…In Sect. 1, for strict partitions µ ⊆ (n) d with d parts, we study generalized flag bundles F µ (E • ) → X introduced by Kempf and Laksov [KL74] (see (4)), and following ideas of [DP15], we give a Gysin formula along these. We get an expression in the spirit of [Dar15] and [DP15], presenting the push-forward as some specific coefficient of a certain polynomial, depending only on the Segre classes of the reference flag of bundles E • (see Theorem 1.2).…”

“…Then, in Sect. 2, we push further the study of the Gysin map along π : G d (E) → X started in [DP15]. We investigate the combinatorics of partitions with at most d parts and establish several formulas for Littlewood-Richardson coefficients.…”

“…Note that like the approaches of [KL74], [Las74], [AF15], and others, our approach uses the desingularization of Schubert varieties (or degeneracy loci) by chains of projective bundles, and various Gysin formulas for them. In our approach however, we use two new Gysin formulas for flag bundles (Proposition 4.4 in [DP15] and Proposition 1.5) and the reasoning is then based on the comparison of these two formulas. To the best of our knowledge, this method is new.…”