Staggered Sheaves on Partial Flag Varieties

Staggered t-structures are a class of t-structures on derived categories of equivariant coherent sheaves. In this note, we show that the derived category of coherent sheaves on a partial flag variety, equivariant for a Borel subgroup, admits an artinian staggered t-structure. As a consequence, we obtain a basis for its equivariant K-theory consisting of simple staggered sheaves. Let X be a variety over an algebraically closed field, and let G be an algebraic group acting on X with finitely many orbits. Let Coh(X) be the category of G-equivariant coherent sheaves on X, and let D(X) denote its bounded derived category. Staggered sheaves, introduced in [1], are the objects in the heart of a certain t-structure on D(X), generalizing the perverse coherent t-structure [2]. The definition of this t-structure depends on the following data: (1) an s-structure on X (see below); (2) a choice of a Serre–Grothendieck dualizing complex ωX ∈ D(X) [4]; and (3) a perversity, which is an integer-valued function on the set of G-orbits, subject to certain constraints. When the perversity is “strictly monotone and comonotone,” the category of staggered sheaves is particularly nice: every object has finite length, and every simple object arises by applying an intermediate-extension (“IC”) functor to an irreducible vector bundle on a G-orbit. An s-structure on X is a collection of full subcategories ({Coh(X)≤n}, {Coh(X)≥n})n∈Z, satisfying various conditions involving Homand Ext-groups, tensor products, and short exact sequences. The staggered codimension of the closure of an orbit iC : C → X, denoted scodC, is defined to be codimC+n, where n is the unique integer such that iCωX ∈ D(C) is a shift of an object in Coh (C)≤n∩Coh(C)≥n. By [1, Theorem 9.9], a sufficient condition for the existence of a strictly monotone and comonotone perversity is that staggered codimensions of neighboring orbits differ by at least 2. The goal of this note is to establish the existence of a well-behaved staggered category on partial flag varieties, by constructing an s-structure and computing staggered codimensions. As a consequence, we obtain a basis for the equivariant K-theory K(G/P ) consisting of simple staggered sheaves. 1. A gluing theorem for s-structures If X happens to be a single G-orbit, s-structures on X can be described via the equivalence between Coh(X) and the category of finite-dimensional representations of the isotropy group of X. In the general case, however, specifying an s-structure on X directly can be quite arduous. The following “gluing theorem” lets us specify an s-structure on X by specifying one on each G-orbit. Theorem 1.1. For each orbit C ⊂ X, let IC ⊂ OX denote the ideal sheaf corresponding to the closed subscheme iC : C ↪→ X. Suppose each orbit C is endowed with an s-structure, and that iCIC |C ∈ Coh(C)≤−1. There is a unique s-structure on X whose restriction to each orbit is the given s-structure. Proof. This statement is nearly identical to [1, Theorem 10.2]. In that result, the requirement that iCIC |C ∈ Coh (C)≤−1 is replaced by the following two assumptions: (F1) For each orbit C, iCIC |C ∈ Coh (C)≤0. (F2) Each F ∈ Coh(C)≤w admits an extension F1 ∈ Coh(C) whose restriction to any smaller orbit C ′ ⊂ C is in Coh(C )≤w. Condition (F1) is trivially implied by the stronger assumption that iCIC |C ∈ Coh (C)≤−1. It suffices, then, to show that (F2) is implied by it as well. Given F ∈ Coh(C)≤w, let G ∈ Coh(C) be some sheaf such that G|C ' F . Let C ′ ⊂ C r C be a maximal orbit (with respect to the closure partial order) such that iC′G|C′ / ∈ Coh (C )≤w. (If there is no such C ′, then G is the desired extension of F , and there is nothing to prove.) Let v ∈ Z be such that iC′G|C′ ∈ Coh (C )≤v. By assumption, we have v > w. Let G′ = G ⊗ I⊗v−w C′ . Since IC′ |XrC′ is isomorphic to the structure sheaf of X r C ′ , The research of the first author was partially supported by NSF grant DMS-0500873. The research of the second author was partially supported by NSF grant DMS-0606300.