Consider a holomorphic torus action on a possibly non-compact Kähler manifold. We show that the higher cohomology groups appearing in the geometric quantization of the symplectic quotient are isomorphic to the invariant parts of the corresponding cohomology groups of the original manifold. For non-Abelian group actions on compact Kähler manifolds, this result was proved recently by Teleman and by Braverman. Our approach is applying the holomorphic instanton complex to the prequantum line bundles over the symplectic cuts. We also settle a conjecture of Zhang and the present author on the exact sequence of higher cohomology groups in the context of symplectic cutting.
Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{ä}hler manifold are sufficient to glue the desired toric hyperk{ä}hler manifold.
This paper gives a partial desingularisation construction for hyperkähler quotients and a criterion for the surjectivity of an analogue of the Kirwan map to the cohomology of hyperkähler quotients. This criterion is applied to some linear actions on hyperkähler vector spaces.
In this paper, metric reduction in generalized geometry is investigated. We show how the Bismut connections on the quotient manifold are obtained from those on the original manifold. The result facilitates the analysis of generalized K$\ddot{a}$hler reduction, which motivates the concept of metric generalized principal bundles and our approach to construct a family of generalized holomorphic line bundles over $\mathbb{C}P^2$ equipped with some non-trivial generalized K$\ddot{a}$hler structures.
We use quasimap Floer cohomology for varying symplectic quotients to resolve several puzzles regarding displaceability of toric moment fibers. For example, we (i) present a compact Hamiltonian torus action containing an {\em open} subset of non-displaceable orbits and a codimension four singular set, partly answering a question of McDuff, and (ii) determine displaceability for most of the moment fibers of a symplectic ellipsoid.
In this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is iso…
This paper examines the relationship between the symplectic quotient X//G of a Hamiltonian G-manifold X, and the associated symplectic quotient X//T, where T is a maximal torus, in the case in which X//G is a compact manifold or orbifold. The three main results are: a formula expressing the rational cohomology ring of X//G in terms of the rational cohomology ring of X//T; an `integration' formula, which expresses cohomology pairings on X//G in terms of cohomology pairings on X//T; and an index formula, which expresses the indices of elliptic operators on X//G in terms of indices on X//T. (The …
The purpose of this paper is to provide a way to compute the intersection cohomology of the GIT quotient of a nonsingular projective variety. We show that the middle perversity intersection cohomology of the GIT quotient $M//G$ is naturally embedded into the $G$ equivariant cohomology of $M^{ss}$ under an assumption satisfied by many interesting spaces. This embedding enables us to compute the intersection pairing in terms of the cup product structure of the equivariant cohomology ring. We also show that the image is a subspace defined by truncation. Overall, we can compute the intersection co…
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 which semistability is not necessarily the same as stability (although we make some weaker assumptions on the action). We also give formulas for intersection pairings on resolutions of singularities (or more precisely partial resolutions, since orbifold singularities are allowed) of the quotients.
Let $M$ be a proper Hamiltonian $K$-space with proper moment map $μ$. The symplectic quotient $X=μ^{-1}(0)/K$ is in general a singular stratified space. In this paper we first generalize the Kirwan map to this symplectic setting which maps the $K$ equivariant cohomology of $μ^{-1}(0)$ to the middle perversity intersection cohomology $IH^*(X)$. Next, we show that there is a natural right inverse of the Kirwan map, under an assumption coming from the cosupport axiom for the intersection homology sheaf. Furthermore, we can identify the subspace of $H^*_K(μ^{-1}(0))$ which corresponds to $IH^*(X)$…
We provided two explicit formulas for the intersection cohomology (as a graded vector space with pairing) of the symplectic quotient by a circle in terms of the $S^1$ equivariant cohomology of the original symplectic manifold and the fixed point data. The key idea is the construction of a small resolution of the symplectic quotient.
We study pseudoholomorphic curves in symplectic quotients as adiabatic limits of solutions of a system of nonlinear first order elliptic partial differential equations in the ambient symplectic manifold. The symplectic manifold carries a Hamiltonian group action. The equations involve the Cauchy-Riemann operator over a Riemann surface, twisted by a connection, and couple the curvature of the connection with the moment map. Our main theorem asserts that the genus zero invariants of Hamiltonian group actions defined by these equations are related to the genus zero Gromov--Witten invariants of th…
Let $M$ be a symplectic manifold, equipped with a Hamiltonian action of a torus $T$. We give an explicit formula for the rational cohomology ring of the symplectic quotient $M//T$ in terms of the cohomology ring of $M$ and fixed point data. Under some restrictions, our formulas apply to integral cohomology. In certain cases these methods enable us to show that the cohomology of the reduced space is torsion-free.
We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkahler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to quiver varieties, and compute as an example the ordinary and equivariant cohomology rings of a hyperpolygon space.
We introduce a de Rham model for stratified spaces arising from symplectic reduction. It turns out that the reduced symplectic form and its powers give rise to well-defined cohomology classes, even on a singular symplectic quotient.
We investigate the small area limit of the gauged Lagrangian Floer cohomology of Frauenfelder. The resulting cohomology theory, which we call quasimap Floer cohomology, is an obstruction to displaceability of Lagrangians in the symplectic quotient. We use the theory to reproduce the results of Fukaya-Oh-Ohta-Ono and Cho-Oh on non-displaceability of moment fibers of not-necessarily-Fano toric varieties and extend their results to toric orbifolds, without virtual fundamental chains. Finally we describe a conjectural relationship to Floer cohomology in the quotient.
We introduce the_inertial cohomology ring_ NH^*_T(Y) of a stably almost complex manifold carrying an action of a torus T. We show that in the case that Y has a locally free action by T, the inertial cohomology ring is isomorphic to the Chen-Ruan orbifold cohomology ring H_{CR}^*(Y/T) of the quotient orbifold Y/T. For Y a compact Hamiltonian T-space, we extend to orbifold cohomology two techniques that are standard in ordinary cohomology. We show that NH^*_T(Y) has a natural ring surjection onto H_{CR}^*(Y//T), where Y//T is the symplectic reduction of Y by T at a regular value of the moment ma…
Given a Hamiltonian system $ (M,ω, G,μ) $ where $(M,ω)$ is a symplectic manifold, $G$ is a compact connected Lie group acting on $(M,ω)$ with moment map $ μ:M \rightarrow\mathfrak{g}^{*}$, then one may construct the symplectic quotient $(M//G, ω_{red})$ where $M//G := μ^{-1}(0)/G$. Kirwan used the norm-square of the moment map, $|μ|^2$, as a G-equivariant Morse function on $M$ to derive formulas for the rational Betti numbers of $M//G$. A real Hamiltonian system $(M,ω, G,μ, σ, ϕ) $ is a Hamiltonian system along with a pair of involutions $(σ:M \rightarrow M, ϕ:G \rightarrow G) $ satisfying cer…
In their 2007 paper, Jarvis, Kaufmann, and Kimura defined the full orbifold $K$-theory of an orbifold ${\mathfrak X}$, analogous to the Chen-Ruan orbifold cohomology of ${\mathfrak X}$ in that it uses the obstruction bundle as a quantum correction to the multiplicative structure. We give an explicit algorithm for the computation of this orbifold invariant in the case when ${\mathfrak X}$ arises as an abelian symplectic quotient. Our methods are integral $K$-theoretic analogues of those used in the orbifold cohomology case by Goldin, Holm, and Knutson in 2005. We rely on the $K$-theoretic Kirwa…
Let $X$ be a smooth projective variety acted on by a reductive group $G$. Let $L$ be a positive $G$-equivariant line bundle over $X$. We use the Witten deformation of the Dolbeault complex of $L$ to show, that the cohomology of the sheaf of holomorphic sections of the induced bundle on the Mumford quotient of $(X,L)$ is equal to the $G$-invariant part on the cohomology of the sheaf of holomorphic sections of $L$. This result, which was recently proven by C. Teleman by a completely different method, generalizes a theorem of Guillemin and Sternberg, which addressed the global sections. It also s…
Consider the Hamiltonian action of a torus on a compact twisted generalized complex manifold $M$. We first observe that Kirwan injectivity and surjectivity hold for ordinary equivariant cohomology in this setting. Then we prove that these two results hold for the twisted equivariant cohomology as well.
We show how certain topological properties of co-K{ä}hler manifolds derive from those of the Kähler manifolds which construct them. We go beyond Betti number results and describe the cohomology algebra structure of co-Kähler manifolds. As a consequence, we prove that co-Kähler manifolds satisfy the Toral Rank Conjecture: $\dim(H^*(M;\mathbb{Q})) \geq 2^r$, for any $r$-torus $T^r$ which acts almost freely on $M$.
Consider the holomorphic Hamiltonian action of a compact Lie group $K$ on a compact Kähler manifold $M$ with a moment map $Φ: M\rightarrow \mathfrak{k}^*$. Assume that $0$ is a regular value of the moment map. Weitsman raised the question of what we can say about the cohomology of the Kähler quotient $M_0:=Φ^{-1}(0)/K$ if all the ordinary cohomology of $M$ is of type $(p, p)$. In this paper, using the Cartan-Chern-Weil theory we show that in the above context there is a natural surjective Kirwan map from an equivariant version of the Dolbeault cohomology of $M$ onto the Dolbeault cohomology of…
We introduce group actions on polyfolds and polyfold bundles. We prove quotient theorems for polyfolds, when the group action has finite isotropy. We prove that the sc-Fredholm property is preserved under quotient if the base polyfold is infinite dimensional. The quotient construction is the main technical tool in the construction of equivariant fundamental class in [42]. We also analyze the equivariant transversality near the fixed locus in the polyfold setting. In the case of $S^1$-action with fixed locus, we give a sufficient condition for the existence of equivariant transverse perturbatio…
This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology $QH_G(X)$ of a smooth complex projective variety X with the action of a connected complex reductive group $G$ to the orbifold quantum cohomology $QH(X//G)$ of its geometric invariant theory quotient $X//G$, and prove that it intertwines the genus zero gauged Gromov-Witten potential of X with the genus zero Gromov-Witten graph potential of $X//G$.
Given a contact manifold $M_#$ together with a transversal infinitesimal automorphism $ξ$, we show that any local leaf space $M$ for the foliation determined by $ξ$ naturally carries a conformally symplectic (cs-) structure. Then we show that the Rumin complex on $M_#$ descends to a complex of differential operators on $M$, whose cohomology can be computed. Applying this construction locally, one obtains a complex intrinsically associated to any manifold endowed with a cs-structure, which recovers the generalization of the so-called Rumin-Seshadri complex to the conformally symplectic setting.…
