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