Cohomology rings of symplectic quotients by circle actions
Ramin Mohammadalikhani
备注：
The typos found after it was first posted on the archive, have been corrected in this new version. The style of the article has changed to match the one that will be published in the Canadian Journal of Mathematics
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 isolated; however the circle action only needs to be ``formally Hamiltonian''. By identifying the kernel, we obtain the cohomology ring of the symplectic quotient. Next we apply this result to some special cases and in particular to the case of products of two dimensional spheres. We show that the results of Kalkman and Hausmann-Knutson are special cases of our result.
We construct a 6-manifold M which admits a smooth circle action and a symplectic form w such that if w' is another symplectic form on M equivalent to w, then (M,w') does not admit a symplectic circle action.
This paper gives methods for understanding invariants of symplectic quotients. The symplectic quotients considered here are compact symplectic manifolds (or more generally orbifolds), which arise as the symplectic quotients of a symplectic manifold by a compact torus. (A companion paper examines symplectic quotients by a nonabelian group, showing how to reduce to the maximal torus.) Let X be a symplectic manifold, with a Hamiltonian action of a compact torus T. The main topological result of this paper describes an explicit cobordism that exists between a symplectic quotient of X by T, and a c…
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.
Let $M$ be a closed Fano symplectic manifold with a semifree Hamiltonian circle action with isolated maximum. We compute the Gromov width and the Hofer-Zehnder capacity of $M$ using a moment map.
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.
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 $(X,σ,J)$ be a compact Kähler Calabi-Yau manifold equipped with a symplectic circle action. By Frankel's theorem \cite{F}, the action on $X$ is non-Hamiltonian and $X$ does not have any fixed point. In this paper, we will show that a symplectic circle action on a compact non-Kähler symplectic Calabi-Yau manifold may have a fixed point. More precisely, we will show that the symplectic $S^1$-manifold constructed by D. McDuff \cite{McD} has the vanishing first Chern class. This manifold has the Betti numbers $b_1 = 3$, $b_2 = 8$, and $b_3 = 12$. In particular, it does not admit any Kähler str…
We show that if a Lie group acts properly on a co-oriented contact manifold preserving the contact structure, then the contact quotient is topologically a stratified space (in the sense that a neighborhood of a point in the quotient is a product of a disk with a cone on a compact stratified space). As a corollary, we obtain that symplectic quotients for proper Hamiltonian actions are topologically stratified spaces in this strong sense thereby extending and simplifying previous work.
We show that the quotient associated to a quasi-Hamiltonian space has a symplectic structure even when 1 is not a regular value of the momentum map: it is a disjoint union of symplectic manifolds of possibly different dimensions, which generalizes a result of Alekseev, Malkin and Meinrenken. We illustrate this theorem with the example of representation spaces of surface groups. As an intermediary step, we show that the isotropy submanifolds of a quasi-Hamiltonian space are quasi-Hamiltonian spaces themselves.
A manifold obtained by k simultaneous symplectic blow-ups of CP^2 of equal sizes epsilon (where the size of CP^1 \subset CP^2 is one) admits an effective two dimensional torus action if k <= 3 and admits an effective circle action if (k-1)epsilon < 1. We show that these bounds are sharp if 1/epsilon is an integer.
Let $M^4$ be a smooth compact manifold with a free circle action generated by a vector field $X.$ Then for any invariant symplectic form $ω$ on $M$ the contracted form $ι_X ω$ is nondegenerate. Using the map $ω--> ι_X ω$ and the related map to $H^1(M / S^1, R)$ we study the topology of the space $S_{inv}$ of invariant symplectic forms. In particular we give a description of $π_0 S_{inv}(M^4)$ in terms of the unit Thurston ball.
Consider a symplectic circle action on a closed symplectic manifold with non-empty isolated fixed points. Associated to each fixed point, there are well-defined non-zero integers, called weights. We prove that the action is Hamiltonian if the sum of an odd number of weights is never equal to zero (the weights may be taken at different fixed points). Moreover, we show that if $\dim M=6$, or if $\dim M=2n \leq 10$ and each fixed point has weights $\{\pm a_1, \cdots, \pm a_n\}$ for some positive integers $a_i$, it is enough to consider the sum of three weights. As applications, we recover the res…
We present some methods to construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or non-symplectic cyclic isotropy point sets. All such actions are not compatible with any symplectic form.
We complete the classification of Hamiltonian torus and circle actions on symplectic four-dimensional manifolds. Following work of Delzant and Karshon, Hamiltonian circle and 2-torus actions on any fixed simply connected symplectic four-manifold were characterized by Karshon, Kessler and Pinsonnault. What remains is to study the case of Hamiltonian actions on blowups of S^2-bundles over a Riemann surface of positive genus. These do not admit 2-torus actions. In this paper, we characterize Hamiltonian circle actions on them. We then derive combinatorial results on the existence and counting of …
Let $(M,ω_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the minimal (or maximal) fixed component of the action is an isolated point, then $(M,ω_M)$ is $S^1$-equivariant symplectomorphic to some Kähler Fano manifold $(X,ω_X, J)$ with a certain holomorphic $\mathbb{C}^*$-action. We also give a complete list of all such Fano manifolds and describe all semifree $\mathbb{C}^*$-actions on them specifically.
We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.
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 …
We show that recent results of Friedl-Vidussi and Chen imply that a symplectic manifold admits a fixed point free circle action if and only if it admits a symplectic circle action and we give a complete description of the symplectic cone in this case. This then completes the characterisation of symplectic 4-manifolds that admit non-trivial circle actions.
Let $(M, ω)$ be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point set consists of isolated points or surfaces. Assume dim $H^2(M)<3$, in \cite{L}, we defined a certain invariant of such spaces which consists of fixed point data and twist type, and we divided the possible values of these invariants into six ``types''. In this paper, we construct such manifolds with these ``types''. As a consequence, we have a precise list of the values of these invariants.
Let $M$ be a symplectic manifold, equipped with a semifree symplectic circle action with a finite, nonempty fixed point set. We show that the circle action must be Hamiltonian, and $M$ must have the equivariant cohomology and Chern classes of $(P^1)^n$.
The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, we will show that (1) if $(M,ω)$ admits a Hamiltonian $S^1$-action, then there exists an $S^1$-invariant symplectic $2$-sphere $S$ in $(M,ω)$ such that $\langle c_1(M), [S] \rangle > 0$, and (2) if the action is non-Hamiltonian, then there exists an $S^1$-invariant symplectic $2$-torus $T$ in $(M,ω)$ such that $\langle c_1(M), [T] \rangle = 0$. As applications, we will give a very simple proof of the following we…
In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the Kähler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here t…
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…
We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to study cyclic actions.
We introduce the notion of regular symplectomorphism and graded regular symplectomorphism between singular phase spaces. Our main concern is to exhibit examples of unitary torus representations whose symplectic quotients cannot be graded regularly symplectomorphic to the quotient of a symplectic representation of a finite group, while the corresponding GIT quotients are smooth. Additionally, we relate the question of simplicialness of a torus representation to Gaussian elimination.
In this paper, we complete the classification of six-dimensional closed monotone symplectic manifolds admitting semifree Hamiltonian $S^1$-actions. We also show that every such manifold is $S^1$-equivariantly symplectomorphic to some Käahler Fano manifold with a certain holomorphic Hamiltonian circle action.
We consider circle bundles over compact three-manifolds with symplectic total spaces. We show that the base of such a space must be irreducible or the product of the two-sphere with the circle. We then deduce that such a bundle admits a symplectic form if and only if it admits one that is invariant under the circle action in three special cases: namely if the base is Seifert fibered, has vanishing Thurston norm, or if the total space admits a Lefschetz fibration.
Suppose a compact Lie group acts on a polarized complex projective manifold (M,L). Under favorable circumstances, the Hilbert-Mumford quotient for the action of the complexified group may be described as a symplectic quotient (or reduction). This paper addresses some metric questions arising from this identification, by analyzing the relationship between the Szego kernel of the pair (M,L) and that of the quotient.
We prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on a construction of Fernandez, Gray and Morgan. We also prove that the only compact complex surfaces admitting free circle actions with contractible orbits are primary Hopf surfaces.
The content of this unpublished paper is subsumed in successive work of the authors, in particular arXiv:1102.0820, arXiv:1102.0821 and arXiv:1205.2434.
We calculate the Riemann-Roch number of some of the pentagon spaces defined in [Klyachko,Kapovich-Millson,HK1]. Using this, we show that while the regular pentagon space is diffeomorphic to a toric variety, even symplectomorphic to one under arbitrarily small perturbations of its symplectic structure, it does not admit a symplectic circle action. In particular, within the cohomology classes of symplectic structures, the subset admitting a circle action is not closed.
In this article, we construct a genus-$0$ or genus-$1$ positive allowable Lefschetz fibration on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As a byproduct, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a sequence of rational blowdowns from its minimal resolution.
We show how to construct a resolution of symplectic orbifolds obtained as quotients of presymplectic manifolds with a torus action. As a corollary, this allows us to desingularise generic symplectic quotients. Given a manifold with a Hamiltonian action of a compact Lie group, symplectic reduction at a coadjoint orbit which is transverse to the moment map produces a symplectic orbifold. If moreover the points of this coadjoint orbit are regular elements of the Lie coalgebra, that is, their stabiliser is a maximal torus, the result for torus quotients may be applied to obtain a desingularisation…
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 …
Let $K$ be a compact Lie group of positive dimension. We show that for most unitary $K$-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group). When $K$ is connected, we show that even a symplectomorphism to a linear symplectic orbifold does not exist. Our results yield conditions that preclude the symplectic quotient of a Hamiltonian $K$-manifold from being locally isomorphic to an orbifold. As an application, we determine which unitary $\operatorname{SU}_2$-modules yield symplectic q…
We derive a complete asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions on arbitrary symplectic manifolds, characterizing the coefficients in the expansion as integrals over the symplectic strata of the corresponding Marsden-Weinstein reduced space and distributions on the Lie algebra. The obtained coefficients involve singular contributions of the lower-dimensional strata related to numerical invariants of the fixed-point set.
We use symplectic cobordism, and the localization result of Ginzburg, Guillemin, and Karshon, to find a wall-crossing formula for the signature of regular symplectic quotients of Hamiltonian torus actions. The formula is recursive, depending ultimately on fixed point data. In the case of a circle action, we obtain a formula for the signature of singular quotients as well. We also show how formulas for the Poincare polynomial and the Euler characteristic (equivalent to those of Kirwan) can be expressed in the same recursive manner.
We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4 and CP^2-bar are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang-Kliener's theorem that only S^4 and CP^2 admit circle-invariant metrics of positive sectional curvature.
We study the existence of symplectic resolutions of quotient singularities V/G where V is a symplectic vector space and G acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the form $K \rtimes S_2$ where $K < \SL_2(\C)$, for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for $\dim V \neq 4$, we classify all quotient singularities $V/G$ admitting a projective symplectic resolution which do not decompose as a product of smaller-dimensional quotient singularities, except for …
Assume $(M, ω)$ is a connected, compact 6 dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict attention to the case $\dim$ $H^2(M)<3$. We give a complete list of the possible manifolds, determine their equivariant cohomology ring and equivariant Chern classes. Some of these manifolds are classified up to diffeomorphism. We also show the existence for a few cases.
We prove that when $d>2$, a $d$-dimensional symplectic quotient at the zero level of a unitary circle representation $V$ such that $V^{\Sp^1}=\{0\}$ cannot be $\Z$-graded regularly symplectomorphic to the quotient of a unitary representations of a finite group.
A generalized moment map is proposed for arbitrary symplectic actions of compact connected Lie groups on closed symplectic manifolds, in the spirit of the circle -valued maps introduced by D. McDuff in the case of non-Hamiltonian circle actions. We study equivariance properties of generalized moments, show that they allow reduction procedures, and obtain in the torus case a version of the Atiyah-Guillemin-Sternberg convexity theorem. As illustration, we reformulate a proof of M.K. Kim that "complexity one" symplectic torus actions are Hamiltonian, and give a symplectic proof of the finiteness …
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.
Let $M$ be a closed 4-manifold with a free circle action. If the orbit manifold $N^3$ satisfies an appropriate fibering condition, then we show how to represent a cone in $H^2(M;\R)$ by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández-Gray-Morgan. In the case that $M$ is the product 4-manifold $S^1\times N$ our construction complements the results of \cite{FV08} (arXiv:0805:1234 [math.GT]) and allows us to completely determine the symplectic cone of such 4-manifolds. The content of this paper partly overlaps with the content of the unpublished prepr…
The symplectic implosion construction of Guillemin, Jeffrey and Sjamaar associates to a Hamiltonian action of a compact group K on a symplectic manifold X its 'imploded cross section'. When X is a complex projective variety and K acts linearly on X, this construction is closely related to geometric invariant theory (GIT) for the action on X of a maximal unipotent subgroup U of the complexification G of K. The aim of this paper is to generalise symplectic implosion to give a symplectic construction for GIT-like quotients by unipotent radicals U of arbitrary parabolic subgroups P of the complex …
Let $K$ be a compact group. For a symplectic quotient $M_λ$ of a compact Hamiltonian Kähler $K$-manifold, we show that the induced complex structure on $M_λ$ is locally invariant when the parameter $λ$ varies in $\mathrm{Lie}(K)^*$. To prove such a result, we take two different approaches: (i) by using the complex geometry properties of the symplectic implosion construction; (ii) by investigating the variation of GIT quotients.
Consider a Hamiltonian circle action on a closed $8$-dimensional symplectic manifold $M$ with exactly five fixed points, which is the smallest possible fixed set. In their paper, L. Godinho and S. Sabatini show that if $M$ satisfies an extra "positivity condition" then the isotropy weights at the fixed points of $M$ agree with those of some linear action on $\mathbb{CP}^4$. Therefore, the (equivariant) cohomology rings and the (equivariant) Chern classes of $M$ and $\mathbb{CP}^4$ agree; in particular, $H^*(M;\mathbb{Z}) \simeq \mathbb{Z}[y]/y^5$ and $c(TM) = (1+y)^5$. In this paper, we prove …
Let $(X,ω_X)$ be a derived scheme with a 0-symplectic form and suppose there is a Hamiltonian $G$-action with a moment map for $G$ a reductive group. We prove, under no further assumptions, that symplectic reduction along any coadjoint orbit in the category of derived Artin stacks has a 0-symplectic form.
Let $K$ be a compact Lie group with complexification $G$, and let $V$ be a unitary $K$-module. We consider the real symplectic quotient $M_0$ at level $0$ of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of $M_0$. We show that if $(V, G)$ is $3$-large, a condition that holds generically, then the complex symplectic quotient has symplectic singularities and is graded Gorenstein. This in particular implies that the real symplectic quotient is graded Gorenstein. In the case that $K$ is a torus or $\operatorname{SU}_2$, we sho…
We identify a family of torus representations such that the corresponding singular symplectic quotients at the $0$-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a subfamily of these torus representations, we give an explicit description of each symplectic quotient as a Poisson differential space with global chart as well as a complete classification of the graded regular diffeomorphism and symplectomorphism classes. Finally, we give explicit examples to indicate that symplectic quotients in this class may …
We consider paths of Hamiltonian diffeomorphism preserving a given compact monotone Lagrangian in a symplectic manifold that extend to an $S^1$--Hamiltonian action. We compute the leading term of the associated Lagrangian Seidel element. We show that such paths minimize the Lagrangian Hofer length. Finally we apply these computations to Lagrangian uniruledness and to give a nice presentation of the Quantum cohomology of real lagrangians in Fano symplectic toric manifolds.
Given a closed surface endowed with a volume form, we equip the space of compatible Riemannian structures with the structure of an infinite-dimensional symplectic manifold. We show that the natural action of the group of volume-preserving diffeomorphisms by push-forward has a group-valued momentum map that assigns to a Riemannian metric the canonical bundle. We then deduce that the Teichmüller space and the moduli space of Riemann surfaces can be realized as symplectic orbit reduced spaces.
Let $S^1$ act on a symplectic manifold in a Hamiltonian fashion with momentum map $Ψ$. Fix a value $a$ of $Ψ$. There is a question of whether the symplectic quotient at $a$ is diffeomorphic to the orbit space of some proper Lie group action. We prove under mild assumptions that this only occurs if the symplectic quotient is diffeomorphic to an effective orbifold. This, in turn, only occurs if $a$ is a regular value, or there is at most one positive weight or at most one negative weight.
Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce many identities concerning the weights on the fixed points. In particular, it gives a criterion to detemine whether or not a symplectic circle action with isolated fixed points is Hamiltonian. As applications, we simplify the proofs of some known results related to symplectic circle actions, due to Godinho, Tolman-Weitsman and Pelayo-Tolman, and generalize so…
Let $(M,ω)$ be a ruled symplectic four-manifold. If $(M, ω)$ is rational, then every homologically trivial symplectic cyclic action on $(M,ω)$ is the restriction of a Hamiltonian circle action.
