This is called algebraic ktheory if we wish to generalize this last example to the equivariant setting, we have to be. The equivariant cohomology of weighted projective spaces. Newest equivariantcohomology questions mathoverflow. Equivariant geometry and the cohomology of the moduli space of curves dan edidin abstract. Even in the case of projective space, though, we do. Equivariant cohomology, koszul duality, and the localization theorem article pdf available in inventiones mathematicae 11 november 2003 with 186 reads how we measure reads. Flag varieties, grassmannians coadjoint orbits of complex semisimple lie groups symplectic toric varieties if m is a hamiltonian t space, then dimm. We call a hamiltonian tspace whose moment map satis. The most important examples are smooth projective varieties y carrying a linear taction. 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. Andrew blumberg, equivariant homotopy theory, 2017 pdf, github a standard text is peter may, equivariant homotopy and cohomology theory, cbms regional conference series in mathematics, vol. Equivariant cohomology harrison chen october 25, 20 contents 1 two koszul dual descriptions of equivariant cohomology on a space with a group action 1 2 koszul duality for dg algeberas and formal cohomological models 3 3 the localization theorem, computing equivariant cohomology, examples 7 introduction.
If an algebraic torus t acts on a complex projective algebraic variety x, then the af. For any such egthe quotient space bg gneghas fundamental group gand trivial higher homotopy groups since eg. The quantum orbifold cohomology of weighted projective spaces. Pdf equivariant cohomology, koszul duality, and the. Kinvariant cw decomposition, and 3 any k space whose ordinary cohomology vanishes in odd degrees cf. 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. Now we study the next more general class of homogeneous spaces, grassmannians.
For quotient stacks xg the categorical co homology ring may be identi ed with the equivariant cohomology h g x. Introduction to equivariant cohomology in algebraic geometry. We describe the integral equivariant cohomology ringof a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. We generalize giventals heuristic argument, which relates small quantum cohomology to s 1equivariant floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small jfunction, a generating function for certain genus. We generalize giventals heuristic argument, which relates small quantum cohomology to s 1equivariant floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small jfunction, a generating function for. The equivariant cohomology of ordinary complex projective space, which corresponds to the case of all weights being equal to 1, is wellknown. We also prove that the qvalued singular cohomology of a delignemumford stack is equal to that of its coarse moduli space. If is a w space, the definition of the equivariant cohomology of is very simple. We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise. We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces.
There the cohomology of these spaces is shown to be free over. C associated with the equivariant cohomology is often an arrangement of linearsubspacesofthevectorspaceht. Kirwan 25 is amongst the first to tackle this by computing betti numbers using equivariant morse theory. Flag varieties, grassmannians coadjoint orbits of complex semisimple lie groups symplectic toric varieties if m is a hamiltonian tspace, then dimm. Equivariant cohomology in algebraic geometry 3 proposition 2. The equivariant cohomology of weighted projective space. In the rst section we introduce the preliminary constructions of equivariant stable homotopy theory from 14 and 6. If a face f of p is regarded as a polyhedron in its own right, then we have a natural equivariant inclusion of x 1f into x 1p. 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. We deduce that the ring is a perfect invariant, and prove a chern class formula for weighted projective bundles. Our main result here is an explicit operator description of the equivariant gromovwitten theory of p1. Equivariant cohomology suppose a topological group g acts continuously on a topological space m.
Introduction to equivariant cohomology in algebraic. Kvect gx is known as equivariant topological ktheory. In fact, on any smooth projective variety, the dualising sheaf is precisely the canonical sheaf. The equivariant cohomology of weighted projective space anthony bahri, matthias franz and nigel ray abstract. 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. 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. Browse other questions tagged equivariantcohomology finitegroups rationalhomotopytheory or ask your own question. We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. We conclude with a discussion of chern classes and a few examples. Our main aim is to obtain explicit descriptions of. In order to understand equivariant sheaves better im trying to prove some basic facts from mackey theory using equivariant sheaves. Now suppose that x is a possibly singular complex projective algebraic variety with an algebraic action of a complex torus t c.
Analogues of this description hold for other complexoriented equivariant cohomology theories, as we confirm in the case of homotopical complex cobordism, which is the universal example. It is the usual cohomology of the quotient of 9 by the diagonal action of 9, where is some contractible space on which acts freely. The simplest homogeneous spaces are the projective spaces studied in bo03. Equivariant cohomology peter may department of mathematics university of chicago may 3, 2012 brandeis university may 5, 2012 cornell university collated slides. Featured on meta community and moderator guidelines for escalating issues via new response. We describe the picard group pichogs of invertible objects in the stable homotopy category of gspectrain terms of a suitable class of homotopy representations of g. The equivariant cohomology ring of weighted projective space, math. Equivariant cohomology of finite group actions and. Equivariant cohomology and equivariant intersection theory. Cohomology of quotients in symplectic and algebraic geometry. A slightly different perspective on invariants of the reduced space is to compute its cohomology ring.
A set of gkm spaces consists of many classes in the spaces with torus actions. The roggraded equivariant ordinary cohomology of complex. We identify all equivariant gromovwitten invariants of p1 as vacuum matrix elements of explicit operators acting in the fock space in the in. For example, when gis a circle s1, then es1 is the unit sphere inside c. Equivariant cohomology is a cohomology theory for gspaces. A convenient description of it comes from the theory of toric varieties. Equivariant structure constants for ordinary and weighted. C cohx, the category of coherent sheaves on an algebraic variety x. For example, its not immediately clear that x 1p should. We deduce that the ring is a perfect invariant, and prove a. The result is obtained by combining the equivariant localization.
Kinvariant cw decomposition, and 3 any kspace whose ordinary cohomology vanishes in odd degrees cf. Gx, the category of gequivariant vector bundles on a topological space x. Lecture on equivariant cohomology imperial college london. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. On the spectrum of the equivariant cohomology ring mark goresky and robert macpherson abstract. Cohomology of quotients in symplectic and algebraic geometry frances clare kirwan. Suppose a connected group g acts on the complex projective space pn. 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. Lecture notes on equivariant cohomology matvei libine arxiv. The equivariant cohomology ring of weighted projective space.
Some versions of cohomology arise by dualizing the construction of. The roggraded equivariant ordinary cohomology of complex projective spaces with linear 2p actions l. We compute the integral torusequivariant cohomology ring for weighted projective space for two different torus actions by embedding the cohomology in a sum of. Gneg0which induces an isomorphism on the homotopy groups and hence an isomorphism on the cohomology groups. Cohomology of projective space let us calculate the cohomology of projective space.