Surjectivity and computing kernels
Kirwan proved that the restriction map from the equivariant cohomology
of a Hamiltonian G-space to
the ordinary cohomology of its symplectic reduction is a
surjection. Jeffrey-Kirwan and Tolman-Weitsman have provided
methods for computing the kernel. In this series of projects, I
exploit these ideas to understand the topology of several different
quotients.
- Distinguishing chambers of the
moment polytope
(with Rebecca Goldin and Lisa Jeffrey) J. Sympl. Geom. 2 (2003), no. 1, 109--131.
Preprint math.SG/0209111
The main theorem in this
paper is that two regular values of the moment map are in the same
chamber if and only if the kernels of the Kirwan map are equal.
- Real loci of symplectic
reductions
(with Rebecca Goldin) Trans. AMS
356 (2004), no. 11, 4623--4642.
Preprint math.SG/0302265
We discuss the real loci of symplectic reductions. We prove an
analogue of Kirwan surjectivity and make a kernel computation.
This applies to real loci of toric varieties and weight varieties.
- Kirwan surjectivity for preorbifold
cohomology
(with Rebecca Goldin and Allen Knutson) Cohomological aspects of Hamiltonian group
actions, Mathematisches Forschungsinstitut Oberwolfach Report
no. 20 (2004) 36--39.
Available in PDF
This announces the results in
Orbifold cohomology of torus
quotients
(with Rebecca Goldin and Allen Knutson)
Preprint math.SG/0502429
We define pre-orbifold cohomology, and give a surjection from
pre-orbifold cohomology to orbifold cohomology of a symplectic
reduction. This makes the computation of orbifold cohomology of a
symplectic reduction that is an orbifold quite straight-forward.
I am currently working on three projects in this area. With Reyer
Sjamaar, I am working to understand the real loci of symplectic
reductions by non-abelian groups. With Rebecca Goldin and Allen
Knutson, I am currently continuing the project on orbifold
cohomology. Finally, with Frank Sottile, I am trying to
understand better the ideals that show up as kernels for symplectic
reductions of flag varieties.