With theobald, he will complete a textbook on applicable algebraic geometry. If we seek a characteristic class satisfying z x tdt x. We consider bases for the cohomology space of regular semisimple hessenberg varieties, consisting of the classes that naturally arise from the bialynickibirula decomposition of the hessenberg varieties. We construct shift operators on equivariant symplectic cohomology which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. In studying topological spaces, one often considers continuous maps f. In the second lecture, i discuss one of the most useful aspects of the theory.
Similar, but not entirely analogous, formulas exist in ktheory 3, cobordism. The resulting interplay has proven to be fertile territory, rich with intricate examples in many disciplines, including integrable systems, enumerative. X,o x then perhaps one is led naturally to the todd class. Elliptic cohomology has deep roots in homotopy theory, algebraic geometry, and mathematical physics. Pdf an introduction to equivariant cohomology and homology. C cohx, the category of coherent sheaves on an algebraic variety x.
A generalization of the conner conjecture and topology of stein spaces dominated by cn. Let eg be a cw complex on which gacts from the left freely and discretely. The aim of these notes is to develop a general procedure for computing the rational cohomology of quotients of group actions in algebraic geometry. 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. X there are also equivariant chern classes and equivariant.
Peter crooks, university of toronto generalized equivariant cohomology and strati. Click get books and find your favorite books in the online library. A more complete description and combinatorial model of the equivariant cohomology of grassmanians is in 3, though we are unable to give much exposition here. Structural aspects of group cohomology rings via equivariant cohomology, university of washington algebra seminar, november 2017 the local cohomology modules of group cohomology rings, informal session at invertibility and duality in homotopy theory and derived algebraic geometry conference, university of regensburg, april 2017 other talks on. Suppose one is interested in some aspect of the topology of x. In the cohomology case, the stable basis elements are linear combinations of the closure of the conormal bundles to the schubert cells, and they form a basis for the localized equivariant cohomology of t gb. Equivariant cohomology university of california berkeley nacfe. Equivariant cohomology is functorial for equivariant maps. Heather macbeth september 1, 2017 abstract the goal of this paper to provide a relatively accessible and integrated introduction to the algebraic.
T1 where t1 is the scalar action, and the answer is well known in the theory of toric varieties see 5. Equivariant cohomology in algebraic geometry 3 the second formula can be proved dually. Ruxandra moraru waterloo andet steven rayan toronto peter crooks, university of toronto generalized equivariant cohomology and strati. Topics in algebra equivariant cohomology, spring 2014. Another application of our isomorphism is a natural algebraic formula for the equivariant pushforward. Equivariant elliptic cohomology is, or is supposed to be, an equivariant version of elliptic cohomology, hence a higher chromatic analogue of equivariant ktheory. Introduced by borel in the late 1950s, equivariant cohomology en codes information about how the topology of a space interacts with a group. In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. The equivariant cohomology algebra in question coincides with the equivariant cohomology of cn. Fulton, equivariant cohomology in algebraic geometry, lectures at columbia. Equivariant cohomology, koszul duality, and the localization. Rebecca goldin department of mathematical sciences george mason university schubert calculus arose from 19th century explorations of enumerative geometry, but has evolved into the study of speci c rings associated with homogeneous spaces.
We shall consider linear actions of complex reductive groups on nonsingular complex projective varieties. We report on a new shifted analogue of edge labeled tableaux. Introductory lectures on equivariant cohomology princeton. A primer november 1, 2018 1 the borel construction let gbe a discrete group and let g x. This formula has found many applications, for example, in analysis, topology, symplectic geometry, and algebraic geometry see 2,6,8,12.
Further remarks 21 acknowledgments 22 references 22 1. For example we could replace by where is a space on which the action is free. Ybe a smooth proper morphism of smooth schemes these hypotheses are not optimal. My understanding is that the plan is for these notes to be compiled into a book at some point. The traditional motivation of these from algebraic topology. Introduction to equivariant cohomology in algebraic geometry. Equivariant cohomology and equivariant intersection theory. This perspective embeds the theory of elliptic cohomology and of the tmf spectrum as an application into the general context of higher topos theory and derived algebraic geometry. This means that for every point x2egthere is a neighborhood x2u x. Gx, the category of g equivariant vector bundles on a topological space x. Today, equivariant localization is a basic tool in mathematical physics, with numerous applications. The aim of equivariant cohomology is to provide a cohomology that will be the cohomology of the quotient in the nice case and that will keep some trace of the nontrivial stabilisers.
X there are also equivariant chern classes and equivariant fundamental classes. Tu many invariants in geometry and topology can be computed asintegrals. The borel construction of the equivariant cohomology h g x goes as follows. The workshop will study this paradigm where the invariant is equivariant elliptic cohomology. Introduction to equivariant cohomology in algebraic. We give an explicit combinatorial description of the support of each class, which enables us to compute the symmetric. For quotient stacks xg the categorical cohomology ring may be identi ed with the equivariant cohomology h g x. Independently on what an intelligent proof of the push. On the spectrum of the equivariant cohomology ring school of. Equivariant elliptic cohomology is, or is supposed to be, an equivariant version of elliptic cohomology, hence a higher chromatic analogue of equivariant ktheory as usual in equivariant cohomology, there is a naive version and refinements thereof, and typically it is these refinements that one is really interested in.
Equivariant geometry and the cohomology of the moduli space of curves dan edidin abstract. To read the fulltext of this research, you can request a copy directly from the author. Fulton ring, which is related to the equivariant cohomology of isotropic grassmannians. This is called algebraic ktheory if we wish to generalize this last example to the equivariant setting, we have to be. Proceedings of the 1984 vancouver conference in algebraic geometry ed. We refer to spa66 for the details and basic properties of these constructions, summarizing the most relevant facts below. Cartans more algebraic approach, and conclude with a discussion of localization principles.
Handbook of moduli university of missouri college of. Algebraic geometry of moduli spaces peter crooks generalized. Quite some time passed before algebraic geometers picked up on these ideas, but in the last. Algebraic geometry, curves, abelian varieties, theta divisors, deformation theory, algebraic topology of varieties, mathematical aspects of quantum field theory. More precisely, we gave in 17 a general formalism of schubert calculus for grassmannians consisting of a factorization algebra factl.
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. Raghunathan on his sixtieth birthday, narosa publishing house, 119125 2004. Induction functor in noncommutative equivariant cohomology and dirac cohomology. Jiewon park project suggested and supervised by dr. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. This book gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology.
One sacrifices some geometric intuition in working with cohomology instead of homology, but one gains the advantage of an easily defined ring structure. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology. Suppose that x is a nice space manifold, algebraic variety acted on by a lie group g. There is no overlap between the two courses, beyond the fact that both make use of very di erent avors of equivariant cohomology. Equivariant cohomology in algebraic geometry 5 example 2. Conjecturally, this gives a littlewoodrichardson rule for the structure constants of the d. Many of the cohomology theories which appear in nature can be extended to equivariant cohomology theories. Pdf this paper provides an introduction to equivariant cohomology and. The morse inequalities are obtained by building up x from the strata and using the thom.
One can always achieve this in topology by an appropriate. 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 quantum cohomology of homogeneous spaces. Atiyahbott localization and the integration formula 10 5. In mathematics, equivariant cohomology or borel cohomology is a cohomology theory from algebraic topology which applies to topological spaces with a group action. Laksov advances in mathematics 217 2008 18691888 of equivariant schubert calculus,thebasis theorem, pieris formula and giambellis formula can be obtained from the corresponding results of the general framework by a change of basis. Pdf equivariant quantum cohomology of homogeneous spaces. Shift operators and connections on equivariant symplectic. It can be considered as the boundary point of the moduli space of elliptic curves. Cohomology algebra of schubert varieties associated to kacmoody groups. Introduced by borel in the late 1950s, equivariant. 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. We will discuss the equivariant cohomology of a manifold endowed with the action of a lie group. Ams geometric representation theory and equivariant.
The generalized elliptic curve that is most relevant to quasielliptic cohomology is the tate curve 4. X\to y, and while equivariant topology also considers such maps, there is the additional constraint that each map respects symmetry in both its domain and target space. On the localization formula in equivariant cohomology loring w. K 0y chtdy ch q y we want to give some example applications. The lectures survey some of the main features of equivariant cohomology at an introductory level. If is a w space, the definition of the equivariant cohomology of is very simple. We take a topological view in this note an algebraic view is probably at least partially known using techniques from derived algebraic geometry, though. It can also be deduced geometrically, using grassmann duality. 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.
Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. When a group g acts suitably on a variety x, the equivariant cohomology of x can. Equivariant algebraic geometry january 30, 2015 note. A survey of elliptic cohomology harvard mathematics department. The first part is an overview, including basic definitions and examples. These are lecture notes from the impanga 2010 summer school. Download full equivariant quantum cohomology of homogeneous spaces book or read online anytime anywhere, available in pdf, epub and kindle. This is called algebraic ktheory if we wish to generalize this last example to the equivariant. Some applications to the equivariant intersection cohomology are given. Cohomology of quotients in symplectic and algebraic geometry.
An introduction to equivariant cohomology and the equivariant. The moment map and equivariant cohomology, topology. One way to do that is to force the action to be free. Equivariant cohomologies and kahlers geometry springerlink. Now suppose that x is a possibly singular complex projective algebraic variety with an algebraic action of a complex torus t c. Kvect gx is known as equivariant topological ktheory.
If an algebraic torus t acts on a complex projective algebraic variety x, then the affine scheme spec h. Topics in algebra schubert calculus and quantum cohomology, fall 2012. Contemporary schubert calculus and schubert geometry. Gysin sequences of rational equivariant cohomology that occur every. The lectures survey some of the main features of equivariant cohomology at an. Create free account to access unlimited books, fast download and ads free. Proceedings of the international conference on algebraic groups and arithmetic volume dedicated to m. Equivariant cohomology is a cohomology theory for gspaces. Post doctoral associates and their fields of interest. On the localization formula in equivariant cohomology. The theory may be useful to specialists in representation theory, algebraic geometry or. As usual in equivariant cohomology, there is a naive version and refinements thereof, and typically it is these refinements that one is really interested in.
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