Equivariant cohomology of juggling varieties in rank one

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.

The main result of the paper is a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subva- riety in the torus equivariant integral cohomology ring of the Grassmannian. As a corollary, we obtain an equivariant version of the Giambelli formula. The (torus) equivariant cohomology rings of flag varieties in general and of the Grassmannian in particular have recently attracted much interest. Here we con- sider the equivariant integral cohomology ring of the Grassmannian. Just as the ordinary Schubert classes form a module basis over the ordinary cohomology ring of a point (namely the ring of integers) for the ordinary integral cohomology ring of the Grassmannian, so do the equivariant Schubert classes form a basis over the equivariant cohomology of a point (namely the ordinary cohomology ring of the classifying space of the torus) for the equivariant cohomology ring (this is true for any generalized flag variety of any type, not just the Grassmannian). Again as in the ordinary case, computing the structure constants of the multiplication with respect to this basis is an interesting problem that goes by the name of Schubert calculus. There is a forgetful functor from equivariant cohomology to ordinary cohomology so that results about the former specialize to those about the latter. Knutson-Tao-Woodward (5) and Knutson-Tao (6) show that the structure con- stants, both ordinary and equivariant, count solutions to certain jigsaw puzzles, thereby showing that they are manifestly positive. In the present paper we take a very different route to computing the equivariant structure constants. Namely, we try to extend to the equivariant case the classical approach by means of the Pieri and Giambelli formulas. Recall, from (3, Eq.(10), p.146) for example, that the Gi- ambelli formula expresses an arbitrary Schubert class as a polynomial with integral coefficients in certain Schubert classes—the Chern classes of the tautolog- ical quotient bundle—and that the Pieri formula expresses as a linear combination of the Schubert classes the product of a special Schubert class with an arbitrary Schubert class. Together they can be used to compute the structure constants. We only partially succeed in our attempt: the first of the three theorems of this paper—see §2 below—is an equivariant Giambelli formula that specializes to the ordinary Giambelli formula as in (3, Eq.(10), p.146), but we still do not have a satisfactory equivariant Pieri formula—see, however, §7 below. The derivation in Fulton (2, §14.3) of the Giambelli formula can perhaps be extended to the equi- variant case, but this is not what we do. Instead, we deduce the Giambelli formula from our second theorem which gives a certain closed-form determinantal formula for the restriction to a torus fixed point of an equivariant Schubert class.

Using the language and terminology of relative homological algebra, in particular that of derived functors, we introduce equivariant cohomology over a general Lie–Rinehart algebra and equivariant de Rham cohomology over a locally trivial Lie groupoid in terms of suitably defined monads (also known as triples) and the associated standard constructions. This extends a characterization of equivariant de Rham cohomology in terms of derived functors developed earlier for the special case where the Lie groupoid is an ordinary Lie group, viewed as a Lie groupoid with a single object; in that theory over a Lie group, the ordinary Bott–Dupont–Shulman–Stasheff complex arises as an a posteriori object. We prove that, given a locally trivial Lie groupoid Ω and a smooth Ω-manifold f : M → B Ω over the space B Ω of objects of Ω, the resulting Ω-equivariant de Rham theory of f reduces to the ordinary equivariant de Rham theory of a vertex manifold f −1(q) relative to the vertex group $$\Omega^q_q$$ , for any vertex q in the space B Ω of objects of Ω; this implies that the equivariant de Rham cohomology introduced here coincides with the stack de Rham cohomology of the associated transformation groupoid; thus this stack de Rham cohomology can be characterized as a relative derived functor. We introduce a notion of cone on a Lie–Rinehart algebra and in particular that of cone on a Lie algebroid. This cone is an indispensable tool for the description of the requisite monads.

Equivariant uniform Alexander-Spanier cohomology theory

Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for G L n ( C ) GL_n(\mathbb {C}) using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equivariant and ordinary cohomology rings of Springer fibers. As an application, we identify an explicit collection of (equivariant) Schubert classes whose images in the (equivariant) cohomology ring of a given Springer fiber form a basis.

(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)

Let $A$ be a finite abelian group. The author sets up an algebraic framework for studying $A$-equivariant complex-orientable cohomology theories in terms of a suitable kind of equivariant formal group. He computes the equivariant cohomology of many spaces in these terms, including projective bundles (and associated Gysin maps), Thom spaces, and infinite Grassmannians.

We study the Demazure–Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern–Schwartz–MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K-theory), in any partial flag manifold. Along the way, we advertise many properties of the left and right divided difference operators in cohomology and K-theory and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K-theory, generating Schubert classes and satisfying a Leibniz rule compatible with the quantum product.

Equivariant de Rham cohomology is extended to the infinite-dimensional setting of a loop subgroup acting on a loop group, using Hida supersymmetric Fock space for the Weil algebra and Malliavin test forms on the loop group. The Mathai–Quillen isomorphism (in the BRST formalism of Kalkman) is defined so that the equivalence of various models of the equivariant de Rham cohomology can be established.

In this paper we study the S 1 {S^1} -equivariant de Rham cohomology of infinite dimensional S 1 {S^1} -manifolds. Our main example is the free loop space L X LX where X X is a finite dimensional manifold with the circle acting by rotating loops. We construct a new form of equivariant cohomology h T ∗ h_T^* which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces L X LX . This gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of Witten [16], Atiyah [2], and Bismut [5]. In particular we show, following Atiyah in [2], that the A ^ \hat A -polynomial of X X arises as an equivariant characteristic class, in the theory h T ∗ h_T^* , of the normal bundle to X X , considered as the space of constant loops, in L X LX .

25 years after its origin, we give a brief introduction to and retrospective of GKM theory. Named after Goresky, Kottwitz, and MacPherson, GKM theory provides a combinatorial algorithm to construct torus-equivariant cohomology for suitable spaces X \mathcal {X} with an appropriate action of a torus T T . We sketch the underlying requirements of the theory and provide concrete examples. We then outline three significant contributions of GKM theory: computing bases for equivariant cohomology, finding formulas for products in the equivariant and ordinary cohomology ring, and constructing geometric representations on the equivariant cohomology. The final section summarizes open questions and research directions.

We extend the work of Allday-Franz-Puppe on syzygies in equivariant cohomology from tori to arbitrary compact connected Lie groups G. In particular, we show that for a compact orientable G-manifold X the analogue of the Chang-Skjelbred sequence is exact if and only if the equivariant cohomology of X is reflexive, if and only if the equivariant Poincare pairing for X is perfect. Along the way we establish that the equivariant cohomology modules arising from the orbit filtration of X are Cohen-Macaulay. We allow singular spaces and introduce a Cartan model for their equivariant cohomology. We also develop a criterion for the finiteness of the number of infinitesimal orbit types of a G-manifold.

This chapter investigates two candidates for equivariant cohomology and explains why it settles on the Borel construction, also called Cartan's mixing construction. Let G be a topological group and M a left G-space. The Borel construction mixes the weakly contractible total space of a principal bundle with the G-space M to produce a homotopy quotient of M. Equivariant cohomology is the cohomology of the homotopy quotient. More generally, given a G-space M, Cartan's mixing construction turns a principal bundle with fiber G into a fiber bundle with fiber M. Cartan's mixing construction fits into the Cartan's mixing diagram, a powerful tool for dealing with equivariant cohomology.

Chiral equivariant cohomology I

In this chapter, we shall proceed to investigate the relationship between the geometric structures of a given G-space X and the algebraic structures of its equivariant cohomology H G * (X). From the viewpoint of transformation groups, those structures which are usually summarized as the orbit structure are certainly the most important geometric structures of a given G-space. Hence, it is almost imperative to investigate how much of the orbit structure of a given G-space X can actually be determined from the algebraic structure of its equivariant cohomology H G * (X). To be more precise, let us formulate a few more specific problems as examples: Problem 1. How much of the cohomology structure of the fixed point set, H*(F), is determined by the equivariant cohomology H G * (X)? Problem 2. Is it possible to give a criterion for the existence of fixed points purely in terms of the equivariant cohomology H G * (X)? Problem 3. Suppose F(H, X) = O. How to determine the set of maximal isotropy subgroups, {Hi ⊂ G; maximal among those H with F(H, X) # O} from the algebraic structure of H G * (X)?