Cohomology of quotients in symplectic and algebraic geometry. Now suppose that x is a possibly singular complex projective algebraic variety with an algebraic action of a complex torus t c. These embeddings are the projectivizations of reductive monoids. An introduction to equivariant cohomology and homology 3 also note that there are many things in gkm which we do not discuss at all in this paper. Any help by way of pointing out errors, typos, or clarifications would be much appreciated.
The notion of cohomology relevant in equivariant stable homotopy theory is the flavor of equivariant cohomology see there for details called bredon cohomology. We have an equivalent description of equivariant cohomology using a kind of. To illustate the geometry behind the operation k let us consider the case u 1. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Our primary reference is the book of chrissginzburg 1, chapters 5 and 6. Via this identification we show that for delignemumford quotient stacks this cohomology is rationally isomorphic to the rational cohomology of the coarse moduli. For many purposes in algebraic geometry, the zariski topology on schemes is too coarse. The lectures survey some of the main features of equivariant cohomology at an introductory level. X,o x then perhaps one is led naturally to the todd class. Meinrenken, equivariant cohomology and the maurercartan equation. We show that for quotient stacks the categorical cohomology may be identified with equivariant cohomology. Equivariant cohomology in algebraic geometry william fulton. Equivariant cohomology also enters into david andersons course on ag varieties gp, but the group in question is a torus and the results are in the direction of algebraic geometry and combinatorics.
Lecture on equivariant cohomology imperial college london. This book gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology. Equivariant cohomology in algebraic geometry william fulton eilenberg lectures, columbia university, spring 2007. Equivariant algebraic geometry january 30, 2015 note. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Algebraically, the obvious next step is to introduce the equivariant q construction on exact categories, give the equivariant version of quillens second definition of algebraic ktheory, and prove the equivalence of the two notions. Quite some time passed before algebraic geometers picked up on these ideas, but in the last twenty years, equivariant techniques have found many applications in enumerative.
Suppose a compact lie group g acts on a topological space x continuously. These are lecture notes from the impanga 2010 summer school. April 30, 2011 abstract introduced by borel in the late 1950s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. Derived equivariant algebraic geometry michael hill. In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. My understanding is that the plan is for these notes to be compiled into a book at some point. An introduction to equivariant cohomology and homology 5 given before.
Quite some time passed before algebraic geometers picked up on these ideas, but in the last. Algebraic geometry sheaves nickolas rollick duration. The equivariant characteristic classes are viewed as classes in the periodic cyclic cohomology of the crossed product by using. First defined in the 1950s, it has been introduced into ktheory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. Working in symplectic geometry, kirwan and many others had studied symplectic reductions, namely quotients of a variety by its group action, and had. Equivariant cohomology, koszul duality, and the localization theorem. We learn about grothendieck topologies, in particular the etale site. Introductory lectures on equivariant cohomology pdf. These days i work mainly in algebraic topology, more specifically on equivariant cohomology. The first part is an overview, including basic definitions and examples. Our main aim is to obtain explicit descriptions of. K 0y chtdy ch q y we want to give some example applications. This formula has found many applications, for example, in analysis, topology, symplectic geometry, and algebraic geometry see 2,6,8,12. Introduction to equivariant cohomology in algebraic geometry.
Ybe a smooth proper morphism of smooth schemes these hypotheses are not optimal. We shall consider linear actions of complex reductive groups on nonsingular complex projective varieties. In order to understand equivariant sheaves better im trying to prove some basic facts from mackey theory using equivariant sheaves. Similar, but not entirely analogous, formulas exist in ktheory 3, cobordism. Classically equivariant cohomology is defined as in wikipedia. In this expository article we give a categorical definition of the integral cohomology ring of a stack. Equivariant cohomology and equivariant intersection theory. The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. Introduction to equivariant cohomology in algebraic. References to some of the general theory of dg algebras is in 4 5 6. Equivariant cohomology in algebraic geometry william. What are some good references to learn the foundations of equivariant homotopy theoryalgebraic topology, for someone who has a background in basic homotopy theory and a tad more advanced algebraic.
On the localization formula in equivariant cohomology. Equivariant complex oriented cohomology theory is discussed in the following articles. Homotopy topoi and equivariant elliptic cohomology ideals. Download citation introduction to equivariant cohomology in algebraic geometry impanga 2010 these are lecture notes from the impanga 2010 summer school. Our main result describes their equivariant cohomology in terms of roots, idempotents, and underlying monoid data. In particular, we obtain the categories of gspaces, for a topological group g, and eschemes, for an einfinityring spectrum e, as full topological. If we seek a characteristic class satisfying z x tdt x. Pdf algebraic cycles and equivariant cohomology theories. Newest equivariantcohomology questions mathematics. I work in nonlinear computational geometry, applying ideas from real algebraic geometry and computational algebraic geometry to solve geometric problems, typically in r3.
One can always achieve this in topology by an appropriate. Introduction to equivariant cohomology in algebraic geometry dave anderson january 28, 2011 abstract introduced by borel in the late 1950s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. This workshop, sponsored by aim and the nsf, will explore computations and examples that will help guide the development of the fledgling field of equivariant derived algebraic geometry. Introduced by borel in the late 1950s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. This interplay has been extensively exploited during recent years.
It explains the main ideas behind some of the most striking recent advances in the subject. We say the group action is free if the stabilizer group gx fg 2 gjgx xg of every point x 2 x is the trivial subgroup. Real solutions, applications, and combinatorics frank sottile summary while algebraic geometry is concerned with basic questions about solutions to equa. Equivariant cohomology of nite group actions steve mitchell fall 2011, mwf 11. Out motivation will be to provide a proof of the classical weyl character formula using a localization result. Newest equivariantcohomology questions mathematics stack. Equivariant cohomology, koszul duality, and the localization. The equivariant algebraic index theorem is a formula expressing the trace on the crossed product algebra of a deformation quantization with a group in terms of a pairing with certain equivariant characteristic classes. If is a w space, the definition of the equivariant cohomology of is very simple.
The aim of these notes is to develop a general procedure for computing the rational cohomology of quotients of group actions in algebraic geometry. Although ideas that fit under this rubric have been around for a long time, recent work on the foundations of equivariant stable homotopy theory starting. Algebraic geometry lecture series markus spitzweck. Hamiltonian tspaces let m be a compact symplectic manifold, with symplectic form. In studying topological spaces, one often considers continuous maps. Also, we characterize those embeddings whose equivariant cohomology ring is obtained via restriction to. Introductory lectures on equivariant cohomology princeton. Equivariant cohomology distinguishes toric manifolds. Equivariant cohomology in algebraic geometry william fulton download bok.
Equivariant cohomology in symplectic geometry rebecca goldin cornell unviersity topology festival may 3, 2008 rebecca goldin gmu equivariant cohomology 1 37. Peter crooks, university of toronto generalized equivariant cohomology and strati. Algebraic cycles and equivariant cohomology theories article pdf available in proceedings of the london mathematical society s3733 november 1996 with reads how we measure reads. Introduction to equivariant cohomology in algebraic geometry dave anderson. This has involved line tangents to objects such as spheres, triangles, or line segments, or classifying degenerate con.
Newest equivariantcohomology questions mathoverflow. The serre spectral sequence and serre class theory 237 9. We describe the equivariant cohomology ring of rationally smooth projective embeddings of reductive groups. Group actions on deformation quantizations and an equivariant. I found the following definition in steenrods cohomology operations in the chapter equivariant cohomology. An introduction to equivariant cohomology and arxiv. The rest of this paper provides an introduction to equivariant cohomology following gkm theory. In the second lecture, i discuss one of the most useful aspects of the theory. Representation theories and algebraic geometry springerlink. Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. Andreas kubel, andreas thom, equivariant differential cohomology, transactions of the american mathematical society 2018 arxiv. Introduced by borel in the late 1950s, equivariant cohomology en codes information about how the topology of a space interacts with a group. We also prove that quasitoric manifolds, which can be. The following exercise gives an example of equivariant poincar.
For the topological equivariant ktheory, see topological ktheory in mathematics, the equivariant algebraic ktheory is an algebraic ktheory associated to the category. Algebraic geometry of moduli spaces peter crooks generalized. Blumberg how best should we be making sense of equivariant derived algebraic geometry. Mackey functors, km,ns, and roggraded cohomology 25 6. Cartans more algebraic approach, and conclude with a discussion of localization principles. The goal of these lectures is to give an introduction to equivariant algebraic ktheory. The 12 lectures presented in representation theories and algebraic geometry focus on the very rich and powerful interplay between algebraic geometry and the representation theories of various modern mathematical structures, such as reductive groups, quantum groups, hecke algebras, restricted lie algebras, and their companions. Equivariant cohomology and the cartan model university of toronto. Equivariant derived algebraic geometry american inst. Introduced by borel in the late 1950s, equivariant. Today, equivariant localization is a basic tool in mathematical physics, with numerous applications. Ruxandra moraru waterloo andet steven rayan toronto peter crooks, university of toronto generalized equivariant cohomology and strati.