Kempf–Laksov Schubert classes for even infinitesimal cohomology theories

Abstract

In this paper, we prove a generalization of Kempf-Laksov formula for the degeneracy loci classes in even infinitesimal cohomology theories of the Grassmannian bundle and the Lagrangian Grassmannian bundle.where d i = p if i = p e − 1 for some integer e and a prime p, and otherwise let d i = 1. It is clear from the expression of the formal group law that the formal inverse is given by ⊟u = −u.A key role in our computation is played by the Segre classes S m (E) of vector bundles E, as it was in [11]. The definition we use is the exact analogue of that given by Fulton in [6]. Using Quillen's formula, which for Ω * was established by Vishik in [17], we manage to describe the generating function of Segre classes in terms of Chern classes. This allows us to define the relative Segre classes S m (E − F ) evaluated on each element [E − F ] of the Grothendieck group of vector bundles. Then, the following geometric interpretation is the main ingredient of our derivation of Kempf-Laksov formula. Proposition A (Proposition 3.8). Let E and F vector bundles over a smooth scheme X, respectively of rank e and f . Consider the dual projective bundle P * (E) π → X with the tautological quotient line bundle O(1). Then for each integer s ≥ 0 we haveWe now explain our main result. Let E be a vector bundle of rank n over a smooth quasiprojective variety X. Consider the Grassmann bundle of rank d subbundles Gr d (E) → X and

K-theoretic Chern class formulas for vexillary degeneracy loci

“…In a sense, the K-theory formulas proved here are the most general ones of their kind: a theorem of Bressler and Evens shows that Schubert classes are sensitive to the choice of desingularization in all cohomology theories beyond K-theory [BE]. Several authors have made significant progress in understanding generalized cohomology theories of flag varieties and the related degeneracy loci (see [HM1,HM2,CPZ,CZZ] and references therein). Formulas in this context must necessarily be of a different flavor, however.…”

Segre classes and Damon–Kempf–Laksov formula in algebraic cobordism

“…In the special case A * = CH * the two notions of Segre classes actually coincide and moreover P CH (z, x) = 1, therefore our expression recovers the classical statement since the only non zero coefficient is a (0,...,0) . For a less trivial application involving formal group laws given by polynomials, we refer the reader to [5] in which we described more explicitly the case of infinitesimal cohomology theories.…”

Stability of Bott--Samelson Classes in Algebraic Cobordism

“…For the reader's convenience, we will briefly recall some basic facts about algebraic cobordism and infinitesimal theories. More details on the construction and the properties of Ω * can be found in [16], while a more comprehensive description of I * n is given in [10]. Both Ω * and I *…”

“…Later, the interest shifted to Grassmann and flag bundles (cf. [13], [4], [12], [11], [10], [7], [8], [9]). One of the main difficulty of Schubert calculus in algebraic cobordism is caused by the fact that the fundamental classes of Schubert varieties are not well-defined in general oriented cohomology theories.…”

“…In [12,11], the first and second authors obtained determinant formula of the cobordism push-forward classes of so-called Damon-Kempf-Laksov resolutions, generalizing the classical Damon-Kempf-Laksov determinant formula of Schubert classes. In [10], more explicit formula of Damon-Kempf-Laksov classes were obtained for infinitesimal cohomology. While these resolutions only exist for Schubert varieties associated to vexillary permutations (like for instance Grassmannian elements), their push-forward classes are stable and so is their determinantal formula.…”