arXiv subjects

R. F. Goldin

Publications and source records attributed to R. F. Goldin.

6 recordsLinked to original sources

arXiv2009

Towards Generalizing Schubert Calculus in the Symplectic Category

The main purpose of this article is to extend some of the ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. Given a generic component of the moment map, we define a canonical class α_p in the equivariant cohomology of the manifold M for each fixed point p of M. When they exist, canonical classes form a natural basis of the equivariant cohomology of M; in particular, when M is a flag variety, these classes are the equivariant Schubert classes. We show that the restriction of a canonical class α_p to a fixed point q can be calculated by a rational function which depends only on the value of the moment map, and the restriction of other canonical classes to points of index exactly two higher. Therefore, the structure constants can be calculated by a similar rational function. Our restriction formula is manifestly positive in many cases, including when M is a flag manifold. Finally, we prove the existence of integral canonical classes in the case that M is a GKM manifold the moment map component is index increasing. In this case, our restriction formula specializes to an easily computable rational sum which depends only on the GKM graph.

math.SG↗

arXiv2004

Cohomology pairings on the symplectic reduction of products

Let M be the product of two compact Hamiltonian T-spaces X and Y. We present a formula for evaluating integrals on the symplectic reduction of M by the diagonal T action. At every regular value of the moment map for X x Y, the integral is the convolution of two distributions associated to the symplectic reductions of X by T and of Y by T. Several examples illustrate the computational strength of this relationship. We also prove a linear analogue which can be used to find cohomology pairings on toric orbifolds.

math.SG↗

arXiv2004

Real loci of symplectic reductions

Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action of a compact $n$-dimensional torus $T$. Suppose that $M$ is equipped with an anti-symplectic involution $σ$ compatible with the $T$-action. The real locus of $M$ is the fixed point set $M^σ$ of $σ$. Duistermaat introduced real loci, and extended several theorems of symplectic geometry to real loci. In this paper, we extend another classical result of symplectic geometry to real loci: the Kirwan surjectivity theorem. In addition, we compute the kernel of the real Kirwan map. These results are direct consequences of techniques introduced by Tolman and Weitsman. In some examples, these results allow us to show that a symplectic reduction $M/ /T$ has the same ordinary cohomology as its real locus $(M/ /T)^{σ_{red}}$, with degrees halved. This extends Duistermaat's original result on real loci to a case in which there is not a natural Hamiltonian torus action.

math.SG↗

arXiv2002

Cohomology of symplectic reductions of generic coadjoint orbits

Let mathcal{O}_lambda be a generic coadjoint orbit of a compact semi-simple Lie group K. Weight varieties are the symplectic reductions of mathcal{O}_lambda by the maximal torus T in K. We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of \mathcal{O}_lambda and it makes explicit the dependence on λand a parameter in Lie(T)^*.

math.SG↗

arXiv2002

An effective algorithm for the cohomology ring of symplectic reductions

Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of $κ:H_G^*(M)\to H^*(M//G)$ is generated by a small number of classes $α\in H_G^*(M)$ satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T-equivariant cohomology of the symplectic reduction of M by T. We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan map $κ:H_G^*(M)\to H^*(M//G)$ is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman in the case T=G and allowing us to apply their methods inductively. This result is new even in the case that dim T = 1. We close with a worked example: the cohomology ring of the product of two $\C P^2$s, quotiented by the diagonal 2-torus action.

math.SG↗

arXiv2002

The cohomology ring of weight varieties and polygon spaces

We use a theorem of Tolman and Weitsman to find explicit formulæfor the rational cohomology rings of the symplectic reduction of flag varieties in C^n, or generic coadjoint orbits of SU(n), by (maximal) torus actions. We also calculate the cohomology ring of the moduli space of $n$ points in CP^k, which is isomorphic to the Grassmannian of k-planes in C^n, by realizing it as a degenerate coadjoint orbit.

math.SG↗