Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31), Volume 31

@inproceedings{Kirwan1984CohomologyOQ,
  title={Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31), Volume 31},
  author={Frances Kirwan},
  year={1984}
}
These notes describe a general procedure for calculating the Betti numbers of the projective quotient varieties that geometric invariant theory associates to reductive group actions on nonsingular complex projective varieties. These quotient varieties are interesting in particular because of their relevance to moduli problems in algebraic geometry. The author describes two different approaches to the problem. One is purely algebraic, while the other uses the methods of symplectic geometry and… 
The geometry and topology of quotient varieties
Let X be a nonsingular projective variety with an algebraic action of a complex torus (c*)n. We study in this thesis the symplectic quotients (reduced phase spaces) and the quotients in a more
On cohomology of invariant submanifolds of Hamiltonian actions
In [5] the author proved that if there is a free algebraic circle action on a nonsingular real algebraic variety X then the fundamental class is trivial in any nonsingular projective complexification
Topological Aspects of Chow Quotients
This paper studies the canonical Chow quotient of a smooth projective variety by a reductive algebraic group. The main purpose is to introduce the Perturbation–Translation–Specialization relation
TOPOLOGICAL ASPECTS OF CHOW QUOTIENTS
This paper studies the canonical Chow quotient of a smooth projective variety by a reductive algebraic group. The main purpose is to introduce the Perturbation–Translation–Specialization relation
The Cohomology Ring of Weight Varieties and Polygon Spaces
We use a theorem of S. Tolman and J. Weitsman (The cohomology rings of Abelian symplectic quotients, math. DG/9807173) to find explicit formulae for the rational cohomology rings of the symplectic
Cohomology of symplectic reductions of generic coadjoint orbits
Let O λ be a generic coadjoint orbit of a compact semi-simple Lie group K. Weight varieties are the symplectic reductions of O λ by the maximal torus T in K. We use a theorem of Tolman and Weitsman
Notes on GIT and symplectic reduction for bundles and varieties
These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties,
Intersection Cohomology of Singular Symplectic Quotients
We give a general procedure for the calculation of the intersection Poincaré polynomial of the symplectic quotient M/K, of a symplectic manifold M by a hamiltonian group action of a compact Lie group
Cohomology pairings on singular quotients in geometric invariant theory
In this paper we shall give formulas for the pairings of intersection cohomology classes of complementary dimensions in the intersection cohomology of geometric invariant-theoretic quotients for
Holomorphic slices, symplectic reduction and multiplicities of representations
I prove the existence of slices for an action of a reductive complex Lie group on a K\"ahler manifold at certain orbits, namely those orbits that intersect the zero level set of a momentum map for
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 17 REFERENCES
A decomposition theorem for the integral homology of a variety
This homology basis formula (1) was motivated by a theorem of Frankel [7]. In [5], also motivated by [7], it was shown that (1) is valid for integer coefficients if X is a compact Kaehler manifold
Principles of Algebraic Geometry
A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications
Partial desingularisations of quotients of nonsingular varieties and their Betti numbers
When a reductive group G acts linearly on a nonsingular complex projective variety X one can define a projective "quotient" variety X//G using Mumford's geometric invariant theory. If the condition
Geometric Invariant Theory
“Geometric Invariant Theory” by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged edition appeared in 1982) is the standard reference on applications of invariant theory to
Convexity and Commuting Hamiltonians
The converse was proved by A. Horn [5], so that all points in this convex hull occur as diagonals of some matrix A with the given eigenvalues. Kostant [7] generalized these results to any compact Lie
Instability in invariant theory
Let V be a representation of a reductive group G. A fundamental theorem in geometric invariant theory states that there are enough polynomial functions on V, which are invariant under G, to
Orbits on Linear Algebraic Groups
Let G be a linear algebraic group and let p: G GL(V) be a rational representation of G. When G is linearly reductive, D. Mumford has shown that if a point x C V has 0 in the Zariski-closure cl (Gs x)
Lectures on Algebraic Topology
I Preliminaries on Categories, Abelian Groups, and Homotopy.- x1 Categories and Functors.- x2 Abelian Groups (Exactness, Direct Sums, Free Abelian Groups).- x3 Homotopy.- II Homology of Complexes.-
On the cohomology groups of moduli spaces of vector bundles on curves
A shaft seal of the type including an annular metal case and a flexible non-elastomeric sealing element such as polytetrafluoroethylene is made by molding a synthetic rubber filler ring between the
Ordinary Differential Equations
Foreword to the Classics Edition Preface to the First Edition Preface to the Second Edition Errata I: Preliminaries II: Existence III: Differential In qualities and Uniqueness IV: Linear Differential
...
1
2
...