Lucía Martín-Merchán, Juan Rojo
We give a method to resolve 4-dimensional symplectic orbifolds making use of techniques from complex geometry and gluing of symplectic forms. We provide some examples to which the resolution method applies.
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We present reformulation of Mathieu's result on representing cohomology classes of symplectic manifold with symplectically harmonic forms. We apply it to the case of foliated manifolds with transversally symplectic structure and to symplectic orbifolds. We obtain in particular that such representation is always possible for compact Kähler orbifolds.
Motivated by an attempt to better understand the notion of a symplectic stack, we introduce the notion of a symplectic hopfoid, which should be thought of as the analog of a groupoid in the so-called symplectic category. After reviewing some foundational material on canonical relations and this category, we show that symplectic hopfoids provide a characterization of symplectic double groupoids in these terms. Then, we show how such structures may be used to produce examples of symplectic orbifolds, and conjecture that all symplectic orbifolds arise via a similar construction. The symplectic st…
We construct the symplectic resolution of a symplectic orbifold whose isotropy locus consists of disjoint submanifolds with homogeneous isotropy, that is, all its points have the same isotropy groups.
We extend Donaldson's asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these suborbifolds, that computes their real cohomology up to middle dimension. We also get the hard Lefschetz and formality properties for them, when the ambient manifold satisfies those properties.
In this paper we present a method to obtain resolutions of symplectic orbifolds arising from symplectic reduction of a Hamiltonian S^1-manifold at a regular value. As an application, we show that all isolated cyclic singularities of a symplectic orbifold admit a resolution and that pre-quantisations of symplectic orbifolds are symplectically fillable by a smooth manifold.
Given an integer b and a finitely presented group G we produce a compact symplectic six-manifold with c_1 = 0, b_2 > b, b_3 > b and fundamental group G. In the simply-connected case we can also arrange for b_3 = 0; in particular these examples are not diffeomorphic to Kähler manifolds with c_1 = 0. The construction begins with a certain orientable four-dimensional hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi-Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the re…
Covering Spaces of Symplectic Toric Orbifolds
该文研究辛环面轨形及其丛的覆盖空间，证明这类覆盖是环面有限子群作用的商，给出轨形丛带标记多面体的一般描述，并将结果用于探究带标记射影空间乘积的环面结构数量。
We develop the Gompf fiber connected sum operation for symplectic orbifolds. We use it to construct a symplectic 4-orbifold with $b_1=0$ and containing symplectic surfaces of genus 1 and 2 that are disjoint and span the rational homology. This is used in turn to construct a K-contact Smale-Barden manifold with specified 2-homology that satisfies the known topological constraints with sharper estimates than the examples constructed previously. The manifold can be chosen spin or non-spin.
Automorphisms of Lie groupoids and symplectic reduction on orbifolds
本文研究了李群体的自同构2-群BAut(X)以及其在orbifold上的辛约简，通过Kan纤维化理论建立了2-群作用与李群体的等价关系，并探讨了辛约简的性质和应用。
We generalize symplectic convexity theorems for Hamiltonian actions with proper momentum maps to symplectic actions on orbifolds with mod-$Γ$ proper momentum maps.
We introduce a method to resolve a symplectic orbifold into a smooth symplectic manifold. Then we study how the formality and the Lefschetz property of the symplectic resolution are compared with that of the symplectic orbifold. We also study the formality of the symplectic blow-up of a symplectic orbifold along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture …
Let $(\X,ω)$ be a compact symplectic orbifold. We define $π_1(Ham(\X, ω))$, the fundamental group of the 2-group of Hamiltonian diffeomorphisms of $(\X, ω)$, and construct a group homomorphism from $π_1(Ham(\X, ω))$ to the group $QH_{orb}^*(\X,Λ)^{\times}$ of multiplicatively invertible elements in the orbifold quantum cohomology ring of $(\X, ω)$. This extends the Seidel representation ([Se], [M]) to symplectic orbifolds.
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…
Twisted Sectors for Lagrangian Floer Theory on Symplectic Orbifolds
本文引入二面体扭转扇区，以在辛轨形上构造 Lagrangian Floer 理论，并探讨相关问题。
Hypertoric varieties are hyperkähler analogues of toric varieties, and are constructed as abelian hyperkähler quotients of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold hypertoric varieties are intimately related to the combinatorics of hyperplane arrangements. By developing hyperkähler analogues of symplectic techniques developed by Goldin, Holm, and Knutson, we give an explicit combinatorial description of the Chen-Ruan orbifold cohomology of an orbifold hypertoric variety in terms of the combinatorial data of a rational coorie…
Making use of theory of differentiable stacks, we study symplectic vortex equations over a compact orbifold Riemann surface. We discuss the category of representable morphisms from a compact orbifold Riemann surface to a quotient stack. After that we define symplectic vortex equations over a compact orbifold Riemann surface. We also discuss the moduli space of solutions to the equations for linear actions of the circle group on the complex plane.
We develop differential and symplectic geometry of differentiable Deligne-Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks as symplectic quotients of C^NxBG, where G is a finite non-abelian group.
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 $K$ 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 use Seidel representation for symplectic orbifolds constructed in Tseng and Wang to compute the quantum cohomology ring of a compact symplectic toric orbifold $(\X,ω)$.
In this paper, we study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman-Weitsman's proof of the GKM theorem in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S of T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen-Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen-Ruan cohomology in terms of orbifold fixed point data.
Let $\sf X$ be a symplectic orbifold groupoid with $\sf S$ being a symplectic sub-orbifold groupoid, and $\sf X_{\mathfrak a}$ be the weight-$\mathfrak a$ blowup of $\sf X$ along $\sf S$ with $\sf Z$ being the corresponding exceptional divisor. We show that there is a weighted blowup correspondence between some certain absolute orbifold Gromov--Witten invariants of $\sf X$ relative to $\sf S$ and some certain relative orbifold Gromov--Witten invariants of the pair $(\sf X_{\mathfrak a}|Z)$. As an application, we prove that the symplectic uniruledness of symplectic orbifold groupoids is a weigh…
Classification of Hamiltonian $S^1$-actions on compact symplectic orbifolds with isolated cyclic singular points in dimension four
本研究将Karshon的分类推广至orbifold情形，通过带标号多重图这一组合不变量刻画空间同构类型，并证明含孤立循环奇点的四维紧辛orbifold上的哈密顿$S^1$-作用空间可由极小空间经有限次等变加权膨胀得到。
We prove that every symplectic toric orbifold is a centered reduction of a Cartesian product of weighted projective spaces. A theorem of Abreu and Macarini shows that if the level set of the reduction passes through a non-displaceable set then the image of this set in the reduced space is also non-displaceable. Using this result we show that every symplectic toric orbifold contains a non-displaceable fiber and we identify this fiber.
These notes accompany a lecture about the topology of symplectic (and other) quotients. The aim is two-fold: first to advertise the ease of computation in the symplectic category; and second to give an account of some new computations for weighted projective spaces. We start with a brief exposition of how orbifolds arise in the symplectic category, and discuss the techniques used to understand their topology. We then show how these results can be used to compute the Chen-Ruan orbifold cohomology ring of abelian symplectic reductions. We conclude by comparing the several rings associated to a w…
Symplectic four-manifolds and hyperbolic $π$-orbifolds
该研究构造了具有双曲π-orbifold结构的闭辛四维流形，解决LeBrun猜想，其最简例欧拉示性数为304，可作为双曲120胞腔拼接而成，存在4次分支覆盖到亏格4的两曲面乘积。
A study of certain symplectic $4$-orbifolds with vanishing canonical class is initiated. We show that for any such symplectic $4$-orbifold $X$, there is a canonically constructed symplectic $4$-orbifold $Y$, together with a cyclic orbifold covering $Y\rightarrow X$, such that $Y$ has at most isolated Du Val singularities and a trivial orbifold canonical line bundle. The minimal resolution of $Y$, to be denoted by $\tilde{Y}$, is a symplectic Calabi-Yau $4$-manifold endowed with a natural symplectic finite cyclic action, extending the deck transformations of the orbifold covering $Y\rightarrow …
Let G be a finite group acting on a symplectic complex vector space V. Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by "symplectic reflectionsd"', i.e. symplectomorphisms with fixed space of codimension 2 in V. Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic. We give a symplectic version of Nakamura conjectures.
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.
We associate to each symplectic $4$-orbifold $X$ a canonical smooth symplectic resolution $π: \tilde{X}\rightarrow X$, which can be done equivariantly if $X$ comes with a symplectic $G$-action by a finite group. Moreover, we show that the resolutions of the symplectic $4$-orbifolds $X/G$ and $\tilde{X}/G$ are in the same symplectic birational equivalence class; in fact, the resolution of $\tilde{X}/G$ can be reduced to that of $X/G$ by successively blowing down symplectic $(-1)$-spheres. To any finite symplectic $G$-action on a $4$-manifold $M$, we associate a pair $(M_G,D)$, where $π: M_G\rig…
In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient $$ W_r=\{(x,y,z,t)|xy-z^{2r}+t^2=0 \}/μ_r(a,-a,1,0), r\geq 1, $$ which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let $X$ and $Y$ be two symplectic orbifolds connected by such a flop. We study orbifold Gromov-Witten invariants of exceptional classes on $X$ and $Y$ and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan.
Monodromy of Nikulin orbifolds
本文针对由 K3^[2] 型四维流形辛商终结化产生的 Nikulin orbifold，提出其单值群极大性的新证明。
Let $G$ be a finite group and $\Y$ a $G$-gerbe over an orbifold $\B$. A disconnected orbifold $\hat{\Y}$ and a flat U(1)-gerbe $c$ on $\hat{\Y}$ is canonically constructed from $\Y$. Motivated by a proposal in physics, we study a mathematical duality between the geometry of the $G$-gerbe $\Y$ and the geometry of $\hat{\Y}$ {\em twisted by} $c$. We prove several results verifying this duality in the contexts of noncommutative geometry and symplectic topology. In particular, we prove that the category of sheaves on $\Y$ is equivalent to the category of $c$-twisted sheaves on $\hat{\Y}$. When $\Y…
We prove a gluing theorem for a symplectic vortex on a compact complex curve and a collection of holomorphic sphere bubbles. Using the theorem we show that the moduli space of regular stable symplectic vortices on a fixed curve with varying markings has the structure of a stratified-smooth topological orbifold. In addition, we show that the moduli space has a non-canonical $C^1$-orbifold structure.
A theorem of E.Lerman and S.Tolman, generalizing a result of T.Delzant, states that compact symplectic toric orbifolds are classified by their moment polytopes, together with a positive integer label attached to each of their facets. In this paper we use this result, and the existence of "global" action-angle coordinates, to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold, by means of smooth functions on the corresponding moment polytope. This is equivalent to parametrizing all toric Kahler metrics and generalizes an analogous…
We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk-deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several ex…
The main purpose of this paper is to summarize the basic ingredients, illustrated with examples, of a pseudoholomorphic curve theory for symplectic 4-orbifolds. These are extensions of relevant work of Gromov, McDuff and Taubes on symplectic 4-manifolds concerning pseudoholomorphic curves and Seiberg-Witten theory. They form the technical backbone of the proof that a symplectic s-cobordism of elliptic 3-manifolds (with a canonical contact structure on the boundary) is smoothly a product. One interesting feature of the theory is that existence of pseudoholomorphic curves gives certain restricti…
Let (M,ω,Φ) be a Hamiltonian T-space and let H be a closed Lie subtorus of T. Under some technical hypotheses on the moment map Φ, we prove that there is no additive torsion in the integral full orbifold K-theory of the orbifold symplectic quotient [M//H]. Our main technical tool is an extension to the case of moment map level sets the well-known result that components of the moment map of a Hamiltonian T-space M are Morse-Bott functions on M. As first applications, we conclude that a large class of symplectic toric orbifolds, as well as certain S^1-quotients of GKM spaces, have integral full …
We prove a generalization of Kawai theorem for the case of orbifold Riemann surface. The computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the Teichmüller space to the $\mathrm{PSL}(2,\mathbb{C})$-character variety, which allows to evaluate explicitly the pullback of Goldman symplectic form in the spirit of Riemann bilinear relations. As a corollary, we obtain a generalization of Goldman's theorem that the pullback of Goldman symplectic from on $\mathrm{PSL}(2,\mathbb{R})$-character variety is a symplectic form of the Weil-Petersson metric…
We study Furuta's finite dimensional approximations for the flow lines of the Moment action functional for toric symplectic orbifolds and show their relation to Givental's toric map space.
We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory of the complex projective line with orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of (polynomial) Frobenius manifolds and integrable systems of Hamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy. We then define a Frobenius structure on the spaces of polynomials in three complex variables of the form F(x,y,z)= -xyz+P_1(x)+P_2(y)+P_3(z) which contains as special cases the ones constructed on the space of Laurent polyn…
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…
We give an intrinsic definition of toric symplectic stacks, and show that they are classified by simple convex polytopes equipped with some additional combinatorial data. This generalizes Delzant's classification of toric symplectic manifolds. As an application, we show that any toric symplectic stack can be deformed to an ineffective toric orbifold.
We prove that the orbifold quantum ring is preserved under singular symplectic flops. Hence we verify Ruan's conjecture for this case.
The ``symplectic cut'' construction [Lerman] produces two symplectic orbifolds $C_-$ and $C_+$ from a symplectic manifold $M$ with a Hamiltonian circle action. We compute the rational cohomology ring of $C_+$ in terms of those of $M$ and $C_-$.
The question of what conditions guarantee that a symplectic $S^1$ action is Hamiltonian has been studied for many years. In a 1998 paper, Sue Tolman and Jonathon Weitsman proved that if the action is semifree and has a non-empty set of isolated fixed points then the action is Hamiltonian. Furthermore, in a 2010 paper Cho, Hwang, and Suh proved in the 6-dimensional case that if we have $b_2^+=1$ at a reduced space at a regular level $λ$ of the circle valued moment map, then the action is Hamiltonian. In this paper, we will use this to prove that certain 6-dimensional symplectic actions which ar…
We use the techniques of integration of Poisson manifolds into symplectic Lie groupoids to build symplectic resolutions (= desingularizations) of the closure of a symplectic leaf. More generally, we show how Lie groupoids can be used to lift singularities, in particular when one imposes a compatibility condition with an additional structure given by a multi-vector field.
Consider a compact symplectic sub-orbifold groupoid $\sf S$ of a compact symplectic orbifold groupoid $(\mathsf X,ω)$. Let $\mathsf X_{\mathfrak a}$ be the weight-$\mathfrak a$ blowup of $\sf X$ along $\sf S$, and $\mathsf D_{\mathfrak a}=\mathsf{PN}_{\mathfrak a}$ be the exceptional divisor, where $\sf N$ is the normal bundle of $\sf S$ in $\sf X$. In this paper we show that the absolute orbifold Gromov--Witten theory of $\mathsf X_{\mathfrak a}$ can be effectively and uniquely reconstructed from the absolute orbifold Gromov--Witten theories of $\sf X$, $\sf S$ and $\mathsf D_{\mathfrak a}$, …
We give Chern-Weil definitions of the Maslov indices of bundle pairs over a Riemann surface Σwith boundary, which consists of symplectic vector bundle on Σand a Lagrangian subbundle on \partialΣ as well as its generalization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold interior singularites, via curvature integral, and find also an analogous topological definition in these cases.
We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.
Motivated by physics, we propose two conjectures regarding the cohomology ring of the crepant resolutions of orbifolds and cohomological invariants of K-equivalent manifolds.
A new transversality condition on orbifolds and integer-valued Gromov-Witten type invariants
本研究引入轨形向量丛截面的FOP横截性条件，可定义伪全纯曲线模空间的整虚闭链，为紧致辛流形整数值Gromov-Witten型不变量定义及哈密顿纤维化上同调分裂定理证明提供新工具。
We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety. In particular, let V be a finite-dimensional complex symplectic vector space and G\subset Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the rational cohomology H^*(X) of any smooth symplectic resolution X \to V/G (multiplicative McKay correspondence). We prove further that if G i…
We derive a numerical criterion for J-holomorphic curves in 4-dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results of Hofer-Lizan-Sikorav and Ivashkovich-Shevchishin to allow punctured curves with boundary that generally need not be somewhere injective or immersed. As an application, we combine this with the intersection theory of punctured holomorphic curves to prove that certain geometrically natural moduli spaces are globally smooth orbifolds, consisting generically of embedded curves, plus unbranched multiple covers that fo…
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 investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold $\mathcal{X}$ and that of its toric crepant resolution $Y$ coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relat…
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 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 …
In this note, we describe a a systematic procedure to find toric crepant resolutions of orbifold vertex, and show that the generating series of certain disc invariants of the orbifold vertex can be suitably identified with the generating series of certain disc invariants of its toric crepant resolutions.
In this continuation of [L-Y1] and [L-L-S-Y], we explain how the Azumaya structure on D-branes together with a netted categorical quotient construction produces the same resolution of ADE orbifold singularities as that arises as the vacuum manifold/variety of the supersymmetric quantum field theory on the D-brane probe world-volume, given by Douglas and Moore [D-M] under the string-theory contents and constructed earlier through hyper-Kähler quotients by Kronheimer and Nakajima. This is consistent with the moral behind this project that Azumaya-type structure on D-branes themselves -- stated a…
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