Equivariant cohomology of moduli spaces of genus three curves. 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. Lecture on equivariant cohomology imperial college london. 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. Equivariant cohomology groups are an important tool for studying gspaces. Equivariant cohomology of moduli spaces of genus three. The equivariant cohomology of weighted projective spaces. Equivariant cohomology in algebraic geometry william fulton eilenberg lectures, columbia university, spring 2007.
The space wgr1,n is the wellknown weighted projective space and its integral cohomology are first studied by kawasaki and its equivariant cohomology by bahrifranzray and tymoczko. Our main aim is to obtain explicit descriptions of. Then k acts on the homogeneous space gk by left translation. If an algebraic torus t acts on a complex projective algebraic variety x, then the af. This means that for every point x2egthere is a neighborhood. This partly duplicates and partly extends a recent calculation done by lewis li using different methods.
The quotient space gh includes examples such as grassmannians and flag manifolds. We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise. On the spectrum of the equivariant cohomology ring mark goresky and robert macpherson abstract. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology.
Equivariant cohomology in algebraic geometry william. Equivariant cohomology is cohomology in the presence of and taking into account groupactions and generally. Many topological and geometrical quantities can be expressed as integrals on a manifold. Equivariant cohomology of weighted grassmannians and. Thus if fe denotes the space of smooth sections of e, and s is such a section of e, then the action of x on the moment map and equivariant cohomology 25 s, written xs, is a welldefined new section, and the operator x has the derivation property. We compute the equivariant hypercohomology of the koszul complex associated to an equivariant vector field. 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. Nov 19, 2018 we study the cohomology of the moduli space of genus three curves with level two structure and some related spaces. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. We show that for quotient stacks the categorical cohomology may be identified with equivariant cohomology. We also make the analogous computations for some related. This is particularly interesting, and traditionally considered, for some choice of geometric cohomology, hence cohomology inside an.
In this expository article we give a categorical definition of the integral cohomology ring of a stack. Newest equivariantcohomology questions mathoverflow. The case of the trivial group provides a new approach to the ktheory orientation 3. A primer november 1, 2018 1 the borel construction let gbe a discrete group and let g x. The equivariant cohomology ring of weighted projective space. We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. Pdf the equivariant cohomology of complexity one spaces. Equivariant structure constants for ordinary and weighted. The space wgr1, n is the wellknown weighted projective space and its integral cohomology are first studied by kawasaki and its equivariant cohomology by bahrifranzray 2 and tymoczko. If is a w space, the definition of the equivariant cohomology of is very simple. An introduction to equivariant cohomology and homology 5 given before. Now suppose that x is a possibly singular complex projective algebraic variety with an algebraic action of a complex torus t c. The overflow blog introducing collections on stack overflow for teams. Introduction if a compact lie group g acts on a manifold m, the space mg of orbits of the action is usually a singular space.
In mathematics, equivariant cohomology or borel cohomology is a cohomology theory from algebraic topology which applies to topological spaces with a group action. Citeseerx document details isaac councill, lee giles, pradeep teregowda. They are used to decide questions about the existence of continuous mappings which camlot be settled by examining cohomology groups alone. Introduction to equivariant cohomology in algebraic geometry. Roman bezrukavnikov december 9, 2017 abstract for a projective variety endowed with a torus action, the equivariant cohomology. Any help by way of pointing out errors, typos, or clarifications would be much appreciated. The example above of a circle g s1 acting on m s2 by rotation shows that this is not a good candidate, since the orbit space mg is a closed interval, a contractible space, so that its cohomology is trivial. The borel construction of the equivariant cohomology h g x goes as follows.
We will give a framework for computing generalized t equivariant cohomology on a t space arising as the direct limit of smooth complex projective t. C associated with the equivariant cohomology is often an arrangement of linearsubspacesofthevectorspaceht. Let eg be a cw complex on which gacts from the left freely and discretely. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. Even in the case of projective space, though, we do. The torus t 2 has euler characteristic zero and the projective space rp 2 is not orientable. Quite some time passed before algebraic geometers picked up on these ideas, but in the last. Reading list for equivariant cohomology mathoverflow. Zbases for the equivariant cohomology rings of a smooth toric variety with respect to the dense torus and a partial.
In fact, the theorem remains true for a locally free action. Let xbe a topological space and ka topological group acting on x. T1 twodimensional topological gravity and equivariant cohomology. How is the equivariant cohomology of a space related to the cohomology of the corresponding. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. An introduction to equivariant cohomology and arxiv. T o assemble the equivarian t cohomology of a complexity one space, we will need a slightly more general form of theorem 1. We deduce that the ring is a perfect invariant, and prove a. X denotes the union of the 0 and 1dimensional orbits of t. The roggraded equivariant ordinary cohomology of complex projective spaces with linear 2p actions l. We deduce that the ring is a perfect invariant, and prove a chern class formula for weighted projective bundles. The moment map and equivariant cohomology sciencedirect.
One can work out the equivariant cohomology ring of this action from general theorems in the literature concerning equivariant cohomology 8, chap. We will be mainly interested in the case where k q is the. The equivariant cohomology measures not only the topology of the space, but also the. As an application of the equivariant thorn isomorphism theorem, we give a new calculation of the additive structure of the equivariant cohomology of complex projective space for g zp.
Equivariant cohomology in algebraic geometry william fulton. Equivariant cohomology is a cohomology theory that takes into account the symmetries of a space. Unlike the ordinary integral cohomology, this ring distinguishes among weighted projective spaces. Equivariant cohomology, koszul duality, and the localization. As a corollary, we give a proof of a version of the agt conjecture concerning pure n2 gauge theory for the group sur. Ih g x is a graded module over hbg, the equivariant cohomology ring of a point. Moreover, we compute explicitly the s 1 equivariant cohomology of the space of the based loops on the complex projective space c p m, where the s 1 action is induced by a. A set of gkm spaces consists of many classes in the spaces with torus actions. Icgx, where icgx is the equivariant cohomology complex of x, and egis the classifying space of g. Equivariant cohomology of a complex projective variety. In short, there is no substitute for simple, explicit examples, and that is the subject of this paper.
C this metric is conformally equivalent to the standard. An introduction to equivariant cohomology and the equivariant. Kinvariant cw decomposition, and 3 any k space whose ordinary cohomology vanishes in odd degrees cf. If x is a smooth quasiprojective gequivariant variety, then 1 any gequivariant coherent sheaf is the quotient of a gequivariant locally free sheaf. We describe the integral equivariant cohomology ringof a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. Flag varieties, grassmannians coadjoint orbits of complex semisimple lie groups symplectic toric varieties if m is a hamiltonian t space, then dimm. Lecture notes on equivariant cohomology matvei libine arxiv. Coming back to the invariant riemann metric on m, by the kornlichtenstein theorem cf. Suppose a connected group g acts on the complex projective space pn. Roughly speaking, equivariant formality amounts to the existence of equivariant extension in the equivariant cohomology theory of any element in the ordinary cohomology theory.
The roggraded equivariant ordinary cohomology of complex. Introduction to equivariant cohomology in algebraic geometry dave anderson 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. Consider torus equivariant cohomology of weighted projective spaces wps complex, projective, nonsmooth in general, but with finite singular locus under some easy conditions on the weights. Pdf equivariant geometry and the cohomology of the. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology ring. We also prove a chern class formula for weighted projective bundles. On gkm description of the equivariant cohomology of a ne flag. The borel cohomology of the loop space of a homogeneous space.
Equivariant cohomology distinguishes toric manifolds arxiv. Equivariant cohomology and the cartan model eckhard meinrenken university of toronto 1. The equivariant cohomology of the moduli space in this section, we recall various facts about the equivariant cohomology of the moduli space of rank 2 semistable vector bundles over a riemann surface x of genus g. Towards the equivariant cohomology ring of moduli spaces. On gkm description of the equivariant cohomology of a ne.
Flag varieties, grassmannians coadjoint orbits of complex semisimple lie groups symplectic toric varieties if m is a hamiltonian tspace, then dimm. Suppose a compact lie group g acts on a topological space x continuously. Equivariant cohomology and fixed points of smooth calogero. My understanding is that the plan is for these notes to be compiled into a book at some point. In general the group g g also acts on the coefficients a a, and then an equivariant cocycle is a map c. Equivariant cohomology in algebraic geometry william fulton download bok. Other readers will always be interested in your opinion of the books youve read. Recall that the classifying space of t is an rfold product of the infinite complex projective space. Exact path integrals by equivariant cohomology arxiv vanity.
If x is a cw complex with cells only in even dimensions and r is a ring, then, by an elementary result in cellular cohomology theory, the ordinary. Since a standard model for homotopy quotients is the borel construction, this kind of equivariant cohomology with trivial g gaction on the coefficients is also called borel equivariant cohomology. We conclude section 5 by including the examples of wgr1,n and wgr2,4. The gcell structure of complex projective spaces with linear 2p actions is discussed in section. Via this identification we show that for delignemumford quotient stacks this cohomology is rationally isomorphic to the rational cohomology of the coarse moduli space. We describe the integral equivariant cohomology of a weighted projective space in terms of piecewise polynomials as well as by generators and relations. We study the cohomology of the moduli space of genus three curves with level two structure and some related spaces. Definition of equivariant cohomology mathematics stack exchange. This hypercohomology is conjectured to be the equivariant cohomology ring of the moduli space mop grc 3 of stable maps of degree 3 from genus zero prestable curves into grassmannians.
We assume x is projective, the quasiprojective case follows by standard embed. The notion of cohomology relevant in equivariant stable homotopy theory is the flavor of equivariant cohomology see there for details called bredon cohomology. Classically equivariant cohomology is defined as in wikipedia. When dimm 2dimt, then we say that m is a symplectic toric variety. Equivariant cohomology is a cohomology theory for gspaces. In studying topological spaces, one often considers continuous maps. Introduction in 1 well behaved cases one expects the cohomology of. In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. Equivariant cohomology and equivariant intersection theory. For example, the gaussbonnet theorem expresses the euler characteristic of a compact oriented surface as an integral of the curvature form.
Equivariant cohomology distinguishes toric manifolds. Request pdf mod equivariant cohomology of homogeneous spaces let g,k be a compact lie pair. 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 equivariant cohomology of weighted projective space. The study of equivariant cohomology combines the ordinary cohomology theory with lie. One of our aims in this paper is to make this analogy precise. Equivariant entire cyclic cohomology of a banach algebra in this section, we introduce a cohomology framework generalizing the entire cyclic cohomology of 3 see also 6, 4, 5, 8 to the equivariant context. On the spectrum of the equivariant cohomology ring 5 iv the following sequence of h.
This will be the second of two presentations on generalized equivariant cohomology and strati. I mostly followed gs1, bgv, ab, par2, and there are no original results in these notes. There the cohomology of these spaces is shown to be free over. Twodimensional topological gravity and equivariant cohomology. Equivariant homology and cohomology of groups sciencedirect. We also prove that quasitoric manifolds, which can be. Mod equivariant cohomology of homogeneous spaces request. The tool of equivariant cohomology on a manifold the hirzebruch surface of complex dimension 2 described in example 14 in subsection 7. Sg 1 jun 2010 june 2, 2010 1 introduction these are the lecture notes for the introductory graduate course i taught at yale during spring 2007. I found the following definition in steenrods cohomology operations in the chapter equivariant cohomology. The equivariant cohomology of weighted projective space anthony bahri, matthias franz and nigel ray abstract. N2 the analogy between topological string theory and equivariant cohomology for differentiable actions of the circle group on manifolds has been widely remarked on.