Introduction to equivariant cohomology in algebraic. These embeddings are the projectivizations of reductive monoids. 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. Derived equivariant algebraic geometry michael hill. Algebraic geometry sheaves nickolas rollick duration.
Via this identification we show that for delignemumford quotient stacks this cohomology is rationally isomorphic to the rational cohomology of the coarse moduli. Lecture on equivariant cohomology imperial college london. We learn about grothendieck topologies, in particular the etale site. 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. My understanding is that the plan is for these notes to be compiled into a book at some point. We shall consider linear actions of complex reductive groups on nonsingular complex projective varieties. We describe the equivariant cohomology ring of rationally smooth projective embeddings of reductive groups.
The first part is an overview, including basic definitions and examples. The notion of cohomology relevant in equivariant stable homotopy theory is the flavor of equivariant cohomology see there for details called bredon cohomology. 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. Equivariant cohomology of nite group actions steve mitchell fall 2011, mwf 11. X,o x then perhaps one is led naturally to the todd class. Ruxandra moraru waterloo andet steven rayan toronto peter crooks, university of toronto generalized equivariant cohomology and strati. Representation theories and algebraic geometry springerlink. An introduction to equivariant cohomology and homology 5 given before. For the topological equivariant ktheory, see topological ktheory in mathematics, the equivariant algebraic ktheory is an algebraic ktheory associated to the category. Working in symplectic geometry, kirwan and many others had studied symplectic reductions, namely quotients of a variety by its group action, and had. Introduced by borel in the late 1950s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. Andreas kubel, andreas thom, equivariant differential cohomology, transactions of the american mathematical society 2018 arxiv. 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. The rest of this paper provides an introduction to equivariant cohomology following gkm theory.
We also prove that quasitoric manifolds, which can be. Also, we characterize those embeddings whose equivariant cohomology ring is obtained via restriction to. An algebraic geometer by training, i have done research at the interface of algebraic geometry, topology, and differential geometry, including hodge theory, degeneracy loci, moduli of vector bundles, and equivariant cohomology. Newest equivariantcohomology questions mathematics. If is a w space, the definition of the equivariant cohomology of is very simple. 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 in algebraic geometry william. Equivariant cohomology, koszul duality, and the localization. The goal of these lectures is to give an introduction to equivariant algebraic ktheory. This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. Mackey functors, km,ns, and roggraded cohomology 25 6. Introductory lectures on equivariant cohomology princeton. Our main result describes their equivariant cohomology in terms of roots, idempotents, and underlying monoid data. 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. Equivariant cohomology in algebraic geometry william fulton. I found the following definition in steenrods cohomology operations in the chapter equivariant cohomology. Equivariant cohomology distinguishes toric manifolds.
K 0y chtdy ch q y we want to give some example applications. The serre spectral sequence and serre class theory 237 9. An introduction to equivariant cohomology and arxiv. 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. Hamiltonian tspaces let m be a compact symplectic manifold, with symplectic form. Equivariant cohomology and equivariant intersection theory michel brion this text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by d. Equivariant derived algebraic geometry american inst. 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. Equivariant algebraic geometry tony feng based on lectures of ravi vakil contents disclaimer 2 1. On the localization formula in equivariant cohomology. Newest equivariantcohomology questions mathematics stack. These are lecture notes from the impanga 2010 summer school. Any help by way of pointing out errors, typos, or clarifications would be much appreciated.
Equivariant cohomology in symplectic geometry rebecca goldin cornell unviersity topology festival may 3, 2008 rebecca goldin gmu equivariant cohomology 1 37. Algebraic geometry lecture series markus spitzweck. Introduced by borel in the late 1950s, equivariant cohomology en codes information about how the topology of a space interacts with a group. Equivariant complex oriented cohomology theory is discussed in the following articles. Classically equivariant cohomology is defined as in wikipedia. Group actions on deformation quantizations and an equivariant.
Today, equivariant localization is a basic tool in mathematical physics, with numerous applications. This formula has found many applications, for example, in analysis, topology, symplectic geometry, and algebraic geometry see 2,6,8,12. Our main aim is to obtain explicit descriptions of. Real solutions, applications, and combinatorics frank sottile summary while algebraic geometry is concerned with basic questions about solutions to equa. These days i work mainly in algebraic topology, more specifically on equivariant cohomology. Introduction to equivariant cohomology in algebraic geometry dave anderson. Cartans more algebraic approach, and conclude with a discussion of localization principles. In this expository article we give a categorical definition of the integral cohomology ring of a stack. References to some of the general theory of dg algebras is in 4 5 6. 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. For many purposes in algebraic geometry, the zariski topology on schemes is too coarse. One can always achieve this in topology by an appropriate. Now suppose that x is a possibly singular complex projective algebraic variety with an algebraic action of a complex torus t c. We show that for quotient stacks the categorical cohomology may be identified with equivariant cohomology.
Equivariant cohomology and the cartan model university of toronto. Algebraic geometry of moduli spaces peter crooks generalized. 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. 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. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. The following exercise gives an example of equivariant poincar. It explains the main ideas behind some of the most striking recent advances in the subject. Equivariant algebraic ktheory northeastern university. Peter crooks, university of toronto generalized equivariant cohomology and strati. In studying topological spaces, one often considers continuous maps.
The aim of these notes is to develop a general procedure for computing the rational cohomology of quotients of group actions in algebraic geometry. Equivariant cohomology, koszul duality, and the localization theorem. Equivariant cohomology and equivariant intersection theory. The equivariant characteristic classes are viewed as classes in the periodic cyclic cohomology of the crossed product by using. Homotopy topoi and equivariant elliptic cohomology ideals. Equivariant cohomology in algebraic geometry william fulton download bok. Quite some time passed before algebraic geometers picked up on these ideas, but in the last. This can all be done, and an exposition will appear in benioff 5. If we seek a characteristic class satisfying z x tdt x. 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.
In the second lecture, i discuss one of the most useful aspects of the theory. 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. Blumberg how best should we be making sense of equivariant derived algebraic geometry. Suppose a compact lie group g acts on a topological space x continuously. Ybe a smooth proper morphism of smooth schemes these hypotheses are not optimal. In order to understand equivariant sheaves better im trying to prove some basic facts from mackey theory using equivariant sheaves. To illustate the geometry behind the operation k let us consider the case u 1.
Newest equivariantcohomology questions mathoverflow. Out motivation will be to provide a proof of the classical weyl character formula using a localization result. I work in nonlinear computational geometry, applying ideas from real algebraic geometry and computational algebraic geometry to solve geometric problems, typically in r3. Pdf algebraic cycles and equivariant cohomology theories.
In particular, we obtain the categories of gspaces, for a topological group g, and eschemes, for an einfinityring spectrum e, as full topological. Introduction to equivariant cohomology in algebraic geometry. This interplay has been extensively exploited during recent years. The lectures survey some of the main features of equivariant cohomology at an introductory level. Cohomology of quotients in symplectic and algebraic geometry. Equivariant cohomology in algebraic geometry william fulton eilenberg lectures, columbia university, spring 2007. 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. We have an equivalent description of equivariant cohomology using a kind of. Meinrenken, equivariant cohomology and the maurercartan equation. Download citation introduction to equivariant cohomology in algebraic geometry impanga 2010 these are lecture notes from the impanga 2010 summer school. This book gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology. Similar, but not entirely analogous, formulas exist in ktheory 3, cobordism.