Manifolds with Group Actions and their Quotients

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AbstractAward: DMS-0606869Principal Investigator: Rebecca GoldinOrbifolds are among the simplest type of singular spaces. Theyarise in symplectic geometry as reductions, in algebraic geometry(where they are called Deligne-Mumford stacks) as certain modulispaces, and in topology as quotient spaces. Algebraic invariantsof these spaces, such as stringy Betti numbers, twisted Hodgenumbers, and Chen-Ruan cohomology (also called orbifoldcohomology) have recently gained interest because they are subtleenough to see the singularities, and in some cases describe thecohomology of a (crepant) resolution of singularities. This grantwill explore new methods to compute the Chen-Ruan cohomology ofan orbifold, following work done by the PI and coauthors in thecase that the orbifold is a global quotient by a compact abelianLie group. These methods should produce a new combinatorialfomula for the Chen-Ruan cohomology of a hypertoric variety (ahyperkahler reduction of quaternionic n-space by a compacttorus), a formula for the K-theory of global abelian quotients,and the Chen-Ruan cohomology of non-abelian (Lie group)quotients. We hope to use these methods to discount or provecertain conjectures about or bifolds and crepant resolutions. Ina separate undertaking, the PI is pursuing a generalcombinatorial formula for questions about the equivariantcohomology of manifolds with circle actions and isolated fixedpoints. We have found a topological formula for the restrictionof certain canonical classes to any other fixed point, which hasled to a kind of classification in 6-dimensions. In higherdimensions, we have found a "positive" restriction formula in thecase that the manifold is Kahler and carries an invariantPalais-Smale metric (or has other rather rigid structure, such asbeing a GKM space). This work should lead to combinatorialformulas for the product structure in the equivariant cohomology,a topic of broad interest because it applies to flag manifoldsand toric varieties, among other varieties. In a third project,we propose to investigate different formulae for the structureconstants in Schubert calculus using methods developed on thecalculus of Bott-Samelson manifolds. This should also generalizeto fixed-point restriction formulas in the case of a symmetricgroup (the fixed points of an involution on G) acting on G/B.This research has roots in classical mechanics. Phase space,which consists of the position and momentum of a particle, is anexample of a symplectic space, with an action by the symmetrygroup, also called a Hamiltonian action. Symmetry arises from thefact that the physics is the same for all observers. By"collapsing" a high-dimensional symplectic space by thesymmetry, we can get a smaller space that is sometimes easier towork with. In a broad sense, this grant is concerned withdescribing these spaces with group actions, and theirquotients. It will have impact on certain questions in physicsthat have arisen in string theory, as well as contribute to thediscussion of how "rigid" our physical universe is byvirtue of symmetry.