Smooth components of Springer fibers

The main result of the paper is that the equivariant intersection cohomology of the semi-stable points on a complex projective variety , for the action of a complex reductive group, may be determined from the equivariant intersection cohomology of the semi-stable points for the action of a maximal torus. It extends the work of Brion who considered the smooth case using equivariant cohomology. Equivariant intersection cohomology is a theory due to Brylinski and the second author. As an application, a surprising relation between the intersection cohomology of Chow hypersurfaces is established in the last section of the paper.

(1.1) This paper concerns three aspects of the action of a compact group K on a space X . The ®rst is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality. This enables one to translate facts about equivariant cohomology into facts about its ordinary cohomology, and back. (3) Equivariant derived category. Many of the results in this paper apply not only to equivariant cohomology, but also to equivariant intersection cohomology. The equivariant derived category provides a framework in both of these may be considered simultaneously, as examples of ``equivariant sheaves''. We treat singular spaces on an equal footing with nonsingular ones. Along the way, we give a description of equivariant homology and equivariant intersection homology in terms of equivariant geometric cycles. Most of the themes in this paper have been considered by other authors in some context. In Sect. 1.7 we sketch the precursors that we know about. For most of the constructions in this paper, we consider an action of a compact connected Lie group K on a space X , however for the purposes of the introduction we will take K ˆ S1† to be a torus. Invent. math. 131, 25±83 (1998)

Reducibility of the intersections of components of a Springer fiber

Let G G be a reductive group satisfying the standard hypotheses, with Lie algebra g \mathfrak {g} . For each nilpotent orbit O 0 \mathcal {O}_0 in a Levi subalgebra g 0 \mathfrak {g}_0 we can consider the induced orbit O \mathcal {O} defined by Lusztig and Spaltenstein. We observe that there is a natural closed morphism of relative dimension zero from the Springer fibre over a point of O 0 \mathcal {O}_0 to the Springer fibre over O \mathcal {O} , which induces an injection on the level of irreducible components. When G = G L N G = GL_N the components of Springer fibres were classified by Spaltenstein using standard tableaux. Our main result explains how the Lusztig–Spaltenstein map of Springer fibres can be described combinatorially, using a new associative composition rule for standard tableaux which we call stacking.

Let G be a connected, complex reductive group and B, a Borel subgroup. The homogeneous space G/B is called the (full) flag variety of G; for GL n this is the ordinary variety of flags of subspaces in ℂ n . The properties of this variety have many implications in the study of reductive groups and their representations. For example, all the irreducible finite dimensional G-modules can be obtained as global sections of line bundles on the flag variety. More generally, any of its (G-linearized) line bundles has at most one non-vanishing sheaf-cohomology space, having a natural structure of a simple G-module if G is simply connected. This is part of the content of the Borel-Weil-Bott theorem and can be seen as a geometric interpretation of E. Cartan’s theory of highest weights.

The quantum double Schubert polynomials studied by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, are shown to represent Schubert classes in Kim’s presentation of the equivariant quantum cohomology of the flag variety. Parabolic analogues of quantum double Schubert polynomials are introduced and shown to represent Schubert classes in the equivariant quantum cohomology of partial flag varieties. This establishes a new method for computing equivariant Gromov-Witten invariants for partial flag varieties. For complete flags Anderson and Chen have announced a proof with different methods.

In [19], A. King states the following conjecture: Any smooth complete toric variety has a tilting bundle whose summands are line bundles. The goal of this paper is to prove King’s conjecture for the following types of smooth complete toric varieties: (i) Any d-dimensional smooth complete toric variety with splitting fan. (ii) Any d-dimensional smooth complete toric variety with Picard number ≤2. (iii) The blow up of any smooth complete minimal toric surface at T-invariants points.

We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen–Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation $${{\mathcal F}}$$ of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of $${{\mathcal F}}$$ vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.

Let L ¯ i →X i be a holomorphic line bundle over a compact complex manifold for i=1,2. Let S i denote the associated principal circle-bundle with respect to some hermitian inner product on L ¯ i . We construct complex structures on S=S 1 ×S 2 which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that L ¯ i are equivariant (ℂ * ) n i -bundles satisfying some additional conditions. The linear type complex structures are constructed assuming X i are (generalized) flag varieties and L ¯ i negative ample line bundles over X i . When H 1 (X 1 ;ℝ)=0 and c 1 (L ¯ 1 )∈H 2 (X 1 ;ℂ) is non-zero, the compact manifold S does not admit any symplectic structure and hence it is non-Kähler with respect to any complex structure.We obtain a vanishing theorem for H q (S;𝒪 S ) when X i are projective manifolds, L ¯ i ∨ are very ample and the cone over X i with respect to the projective imbedding defined by L ¯ i ∨ are Cohen-Macaulay. We obtain applications to the Picard group of S. When X i =G i /P i where P i are maximal parabolic subgroups and S is endowed with linear type complex structure with “vanishing unipotent part” we show that the field of meromorphic functions on S is purely transcendental over ℂ.

We introduce the notion of flag Bott-Samelson variety as a generalization of Bott-Samelson variety and flag variety. Using a birational morphism from an appropriate Bott-Samelson variety to a flag Bott-Samelson variety, we compute Newton-Okounkov bodies of flag Bott-Samelson varieties as generalized string polytopes, which are applied to give polyhedral expressions for irreducible decompositions of tensor products of $G$-modules. Furthermore, we show that flag Bott-Samelson varieties are degenerated into flag Bott manifolds with higher rank torus actions, and find the Duistermaat-Heckman measures of the moment map images of flag Bott-Samelson varieties with the torus action together with invariant closed $2$-forms.

We give a K K -theoretic criterion for a quasi-projective variety to be smooth. If L \mathbb {L} is a line bundle corresponding to an ample invertible sheaf on X X , it suffices that K q ( X ) ≅ K q ( L ) K_q(X)\cong K_q(\mathbb {L}) for all q ≤ dim ⁡ ( X ) + 1 q\le \dim (X)+1 .

In this paper we explore nonabelian gauged linear sigma models (GLSMs) for symplectic and orthogonal Grassmannians and flag manifolds, checking e.g. global symmetries, Witten indices, and Calabi-Yau conditions, following up a proposal in the math community. For symplectic Grassmannians, we check that Coulomb branch vacua of the GLSM are consistent with ordinary and equivariant quantum cohomology of the space.

We link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth. Nous relions des variétés de Schubert dans le variété flag généralisée avec des arrangements des hyperplans. Pour un élément dún groupe de Weyl, nous construisons un certain arrangement graphique des hyperplans. Nous montrons que la fonction génératrice pour les régions de cet arrangement coincide avec le polynome de Poincaré de la variété de Schubert correspondante si et seulement si la variété de Schubert est rationnellement lisse.

Let G be a connected reductive complex algebraic group acting on a smooth complete complex algebraic variety X. We assume that X is a regular embedding, a condition satisfied in particular by smooth toric varieties and flag varieties. For any set D of G-stable prime divisors, we study the action on X of the group Aut°(X, D), the connected automorphism group of X stabilizing all elements of D. We determine a Levi subgroup A(X, D) of Aut°(X, D), and also relevant invariants of X as a spherical A(X, D)-variety. As a byproduct, we obtain a complete description of the inclusion relation between closures of A(X, D)-orbits on X.

We prove a local analog of the Deligne-Riemann-Roch isomorphism in the case of line bundles and relative dimension $1$. This local analog consists in computation of the class of $12$th power of the determinant central extension of a group ind-scheme $\mathcal G$ by the multiplicative group scheme over $\mathbb Q$ via the product of $2$-cocyles in the second cohomology group. These $2$-cocycles are the compositions of the Contou-Carrère symbol with the $\cup$-product of $1$-cocycles. The group ind-scheme $\mathcal{G}$ represents the functor which assigns to every commutative ring $A$ the group that is the semidirect product of the group $A((t))^*$ of invertible elements of $A((t))$ and the group of continuous $A$-automorphisms of $A$-algebra $A((t))$. The determinant central extension naturally acts on the determinant line bundle on the moduli stack of geometric data (proper quintets). A proper quintet is a collection of a proper family of curves over $\operatorname{Spec} A$, a line bundle on this family, a section of this family, a relative formal parameter at the section, a formal trivialization of the bundle at the section that satisfy further conditions.