Edidin–Graham

Edidin–Graham
Name	Edidin–Graham
Field	Algebraic geometry
Introduced	1990s
Notable	Dan Edidin; William Graham

Edidin–Graham is a construction and set of techniques in algebraic geometry linking equivariant intersection theory, characteristic classes, and localization methods. It provides tools to compute Chow groups, equivariant Chow rings, and generalized Chern classes for actions of algebraic groups on schemes and stacks, and it connects foundational work in equivariant cohomology with computational approaches in intersection theory. The development built on interactions among researchers in algebraic geometry, representation theory, and topology and has been applied to problems arising in moduli theory, enumerative geometry, and equivariant K-theory.

Definition and Overview

The Edidin–Graham framework formalizes an equivariant intersection theory for algebraic group actions by combining ideas from intersection theory of William Fulton's algebraic cycles, equivariant cohomology inspired by Raoul Bott and Raoul Bott and Atiyah-style localization, and classifying-space techniques reminiscent of constructions by Quillen and Milnor. In this setting one replaces a quotient by an algebraic approximation to a classifying space such as a principal bundle associated to GL_n or other linear algebraic groups like SL_n, PGL_n, Torus, and uses mixed quotients to define equivariant Chow groups and equivariant Chern classes. The approach is directly tied to methods in the work of Grothendieck on characteristic classes, the machinery of Intersection theory, and computational advances following ideas from Atiyah–Bott localization and Borel's equivariant cohomology.

Historical Development

Origins of the Edidin–Graham methods trace to late 20th-century efforts to create an algebraic analogue of topological equivariant cohomology, drawing on prior foundational contributions by Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, and later formalizations by William Fulton and Robert MacPherson. The specific formulation emerged in research programs of the 1990s and early 2000s when Dan Edidin and William Graham published influential papers adapting classifying-space approximations to define equivariant Chow groups and operational Chern classes for quotient stacks and moduli spaces. Their work interacted with contemporaneous results from researchers such as Maxim Kontsevich, Andrei Okounkov, and Nigel Hitchin in moduli problems and with developments by Graham and Kumar and others in Schubert calculus. Subsequent expansions incorporated techniques from Kresch on intersection theory for stacks, and from Brion on linearization of group actions.

Mathematical Formulation

At its core, the Edidin–Graham formalism defines equivariant Chow groups A^G_*(X) for a scheme X with action by a linear algebraic group G by choosing a representation V of G with open subset U on which G acts freely, and forming the mixed quotient (X × U)/G. This mirrors the Borel construction used by Armand Borel in equivariant cohomology and echoes classifying-space constructions by D. Quillen and J. Milnor. The equivariant Chern classes are obtained as operational classes in the ring A_G^*(X), compatible with pullback and pushforward maps analogous to those in Fulton's intersection theory. Localization theorems in this context relate A^G_*(X) to contributions from fixed-point loci under subtori such as G_m and maximal tori of GL_n; these results parallel the localization formulae of Atiyah–Bott and Berline–Vergne and use weights arising from representations of reductive groups like SO_n and Sp_n. The formalism also interfaces with equivariant K-theory approaches due to Graham and Kumar and localization in equivariant K-theory developed by Atiyah and Segal.

Applications and Examples

Edidin–Graham techniques have been applied to compute equivariant Chow rings of flag varieties and Grassmannians associated to GL_n and to analyze intersection theory on quotient stacks such as moduli stacks of vector bundles over curves studied by Nigel Hitchin and Carlos Simpson. They provide computational tools for Schubert calculus problems connected to H. Weyl character formulas and to enumerative results in Gromov–Witten theory developed by Kontsevich and Manin. Examples include explicit calculations for torus actions on projective spaces P^n and for actions on Hilbert schemes related to work by Nakajima; applications also extend to degeneracy loci computations used in proofs involving theorems of Thom-Porteous type and to analyses of Chern-Schwartz-MacPherson classes building on Robert MacPherson's work. In moduli theory, the methods aid in intersection calculations on stacks appearing in geometric invariant theory developed by David Mumford and in studying tautological rings on moduli spaces of curves related to Deligne–Mumford stacks.

Related Theorems and Extensions

The Edidin–Graham construction connects to several major results and extensions in algebraic geometry and representation theory. It complements the equivariant Riemann–Roch theorem as formulated by Fulton and Gillet and relates to the Beilinson–Bernstein localization paradigm in representation theory influenced by Joseph Bernstein and Alexander Beilinson. Extensions include adaptations to algebraic stacks by Kresch and to motivic and K-theoretic settings building on work by Thomason and Levine. Localization theorems in the Edidin–Graham spirit align with virtual localization techniques of Graber and Pandharipande, and interactions with equivariant derived categories reflect developments by Bondal and Van den Bergh.

Notable Contributors and Influence

Key contributors include Dan Edidin and William Graham who formulated and popularized the framework; subsequent influential work was carried out by Andrei Kresch, Michel Brion, William Fulton, Barbara Fantechi, Eduardo Esteves, and Rahul Pandharipande among others. The approach has influenced research programs in intersection theory, moduli spaces, Schubert calculus, and equivariant K-theory, shaping directions pursued by researchers such as Maxim Kontsevich, Andrei Okounkov, Nikita Nekrasov, and Bertram Kostant. The methods are now standard tools in the toolkit of algebraic geometers working on problems involving group actions, quotient stacks, and enumerative geometry.

Category:Algebraic geometry