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Register a new userEquivariant Cohomology of Infinite-Dimensional Grassmannian and Shifted Schur Functions
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We study the multiplication and comultiplication in equivariant cohomology of Sato Grassmannian
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We investigate the equivariant intersection cohomology of a toric variety. Considering the defining fan of the variety as a finite topological space with the subfans being the open sets (that corresponds to the toric topology given by the invariant open subsets), equivariant intersection cohomology provides a sheaf (of graded modules over a sheaf of graded rings) on that fan space. We prove that this sheaf is a minimal extension sheaf, i.e., that it satisfies three relatively simple axioms which are known to characterize such a sheaf up to isomorphism. In the verification of the second of these axioms, a key role is played by equivariantly formal toric varieties, where equivariant and usual (non-equivariant) intersection cohomology determine each other by Kunneth type formulae. Minimal extension sheaves can be constructed in a purely formal way and thus also exist for non-rational fans. As a consequence, we can extend the notion of an equivariantly formal fan even to this general setup. In this way, it will be possible to introduce virtual intersection cohomology for equivariantly formal non-rational fans.
We introduce a Bredon motivic cohomology theory for smooth schemes defined over a field and equipped with an action by a finite group. These cohomology groups are defined for finite dimensional representations as the hypercohomology of complexes of equivariant correspondences in the equivariant Nisnevich topology. We generalize the theory of presheaves with transfers to the equivariant setting and prove a Cancellation Theorem.
This paper provides an introduction to equivariant cohomology and homology using the approach of Goresky, Kottwitz, and MacPherson. When a group G acts suitably on a variety X, the equivariant cohomology of X can be computed using the combinatorial data of a skeleton of G-orbits on X. We give both a geometric definition and the traditional definition of equivariant cohomology. We include a discussion of the moment map and an algorithm for finding a set of generators for the equivariant cohomology of X. Many examples and explicit calculations are provided.
We show a Z^2-filtered algebraic structure and a quantum to classical principle on the torus-equivariant quantum cohomology of a complete flag variety of general Lie type, generalizing earlier works of Leung and the second author. We also provide various applications on equivariant quantum Schubert calculus, including an equivariant quantum Pieri rule for any partial flag variety of Lie type A.
A general vanishing result for the first cohomology group of affine smooth complex varieties with values in rank one local systems is established. This is applied to the determination of the monodromy action on the first cohomology group of the Milnor fiber of some line arrangements, including the monomial arrangement and the exceptional reflection arrangement of type $G_{31}$.
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