Complete invariants for Hamiltonian torus actions with two dimensional quotients

2-环面辛商的 Hilbert 级数

Hilbert series of symplectic quotients by the 2-torus

本文研究 2-环面线性辛商的实正则函数分次代数，给出其 Hilbert 级数及在 t=1 处 Laurent 展开前四个系数的计算方法，并提供可用于具体例子的算法。

环面作用的凯勒商的比较

Comparison of Kähler quotients of torus actions

研究环面作用下紧致凯勒哈密顿\(T\)-流形的凯勒商，利用几何不变理论变分理论，证明\(\alpha\)变化时商\(X_{\alpha}\)经历双有理变换，可比较凯勒类，还得到奇异凯勒商的部分有理去奇异化及黎曼 - 罗赫数公式。

Transversality theory, cobordisms, and invariants of symplectic quotients

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…

On Orbifold Criteria for Symplectic Toric Quotients

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.

Weak splittings of quotients of Drinfeld and Heisenberg doubles

We investigate the fine structure of the simplectic foliations of Poisson homogeneous spaces. Two general results are proved for weak splittings of surjective Poisson submersions from Heisenberg and Drinfeld doubles. The implications of these results are that the torus orbits of symplectic leaves of the quotients can be explicitly realized as Poisson-Dirac submanifolds of the torus orbits of the doubles. The results have a wide range of applications to many families of real and complex Poisson structures on flag varieties. Their torus orbits of leaves recover important families of varieties su…

Integral Geometry and Hamiltonian volume minimizing property of a totally geodesic Lagrangian torus in S^2 times S^2

We prove that the product of equators $S^{1} \times S^{1}$ in $S^{2} \times S^{2}$ is globally volume minimizing under Hamiltonian deformations.

Almost all Lagrangian torus orbits in ${\mathbb C}P^n$ are not Hamiltonian volume minimizing

All principal orbits of the standard Hamiltonian $T^n$-action on the complex projective space ${\mathbb C}P^n$ are Lagrangian tori.In this article, we prove that most of them are not volume minimizing under Hamiltonian isotopies of ${\mathbb C}P^n$ if the complex dimension $n$ is greater than two, although they are Hamiltonian minimal and Hamiltonian stable.

Toeplitz operators and Hamiltonian torus action

This paper is devoted to semi-classical aspects of symplectic reduction. Consider a compact prequantizable Kahler manifold M with a Hamiltonian torus action. Guillemin and Sternberg introduced an isomorphism between the invariant part of the quantum space associated to M and the quantum space associated to the symplectic quotient of M, provided this quotient is non-singular. We prove that this isomorphism is a Fourier integral operator and that the Toeplitz operators of M descend to Toeplitz operators of the reduced phase space. We also extend these results to the case where the symplectic quo…

On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide explicit examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct quasimorphisms on $Ham(T^2)$ and some of them are Calabi.

Continuous families of Hamiltonian torus actions

We determine conditions under which two Hamiltonian torus actions on a symplectic manifold $M$ are homotopic by a family of Hamiltonian torus actions, when $M$ is a toric manifold and when $M$ is a coadjoint orbit.

Orbifold cohomology of torus quotients

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…

辛商空间的Gelfand-Cetlin阿贝尔化

Gelfand-Cetlin abelianizations of symplectic quotients

本文证明紧连通李群作用的哈密顿G-空间的一般辛商空间可等价地由紧环面作用得到，该环面作用源于g*上的可积系统，从而实现了辛商空间的阿贝尔化。

Topology of generalized complex quotients

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.

All the GIT quotients at once

Let T be an algebraic torus acting on a smooth variety V. We study the relationship between the various GIT quotients of V and the symplectic quotient of the cotangent bundle of V.

Quasi-Hamiltonian quotients as disjoint unions of symplectic manifolds

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.

Convexity for Hamiltonian torus actions on $b$-symplectic manifolds

In [GMPS] we proved that the moment map image of a $b$-symplectic toric manifold is a convex $b$-polytope. In this paper we obtain convexity results for the more general case of non-toric hamiltonian torus actions on $b$-symplectic manifolds. The modular weights of the action on the connected components of the exceptional hypersurface play a fundamental role: either they are all zero and the moment map behaves as in classic symplectic one, or they are all nonzero and the moment map behaves as in the toric $b$-symplectic case studied in [GMPS].

A Wall-crossing Formula for the Signature of Symplectic Quotients

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.

Darboux transforms and spectral curves of Hamiltonian stationary Lagrangian tori

The multiplier spectral curve of a conformal torus in the 4-sphere is essentially, see arXiv:0712.2311, given by all Darboux transforms of the conformal torus. In the particular case when the conformal immersion is a Hamiltonian stationary torus in Euclidean 4-space, the left normal of the immersion is harmonic, hence we can associate a second Riemann surface: the eigenline spectral curve of the left normal, as defined by Hitchin. We show that the multiplier spectral curve of a Hamiltonian stationary torus and the eigenline spectral curve of its left normal are biholomorphic Riemann surfaces o…

Hard Lefschetz Property for Hamiltonian torus actions on 6-dimensional GKM manifolds

In this paper, we study the hard Lefschetz property of a symplectic manifold which admits a Hamiltonian torus action. More precisely, let $(M,ω)$ be a 6-dimensional compact symplectic manifold with a Hamiltonian $T^2$-action. We will show that if the moment map image of $M$ is a GKM-graph and if the graph is index-increasing, then $(M,ω)$ satisfies the hard Lefschetz property.

全局商空间的轨道哈密顿Floer理论

Orbifold Hamiltonian Floer theory for global quotients

针对光滑闭辛流形被有限群忠实辛作用商化得到的全局商轨道，构建了体形变轨道哈密顿Floer理论，通过导出轨道形为模空间赋予一致表示。

Centered complexity one Hamiltonian torus actions

We consider symplectic manifolds with Hamiltonian torus actions which are "almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are "centered" and the moment map is proper. In particular, this classifies the moment map preimages of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmanians Gr^+(2,R^5) and Gr^+(2,R^6) by two equal symplectic balls.

Equivariant fixed point formulae and Toeplitz operators under Hamiltonian torus actions

The main result of this paper is the description of asymptotics along rays in weight space of traces of equivariant Toeplitz operators composed with quantomorphisms for torus actions. The main ingredient in the proof is the microlocal analysis of the equivariant Szego kernels.

Equivariant Metaplectic-c Prequantization of Symplectic Manifolds with Hamiltonian Torus Actions

This paper determines a condition that is necessary and sufficient for a metaplectic-c prequantizable symplectic manifold with an effective Hamiltonian torus action to admit an equivariant metaplectic-c prequantization. The condition is evaluated at a fixed point of the momentum map, and is shifted from the one that is known for equivariant prequantization line bundles. Given a metaplectic-c prequantized symplectic manifold with a Hamiltonian energy function, the author previously proposed a condition under which a regular value of the function should be considered a quantized energy level of …

Kirwan surjectivity in K-theory for Hamiltonian loop group quotients

Let G be a compact Lie group and LG its associated loop group. The main result of this manuscript is a surjectivity theorem from the equivariant K-theory of a Hamiltonian LG-space onto the integral K-theory of its Hamiltonian LG-quotient. Our result is a K-theoretic analogue of previous work in rational Borel-equivariant cohomology of Bott, Tolman, and Weitsman. Our proof techniques differ from that of Bott, Tolman, and Weitsman in that they explicitly use the Borel construction, which we do not have at our disposal in equivariant K-theory; we instead directly construct G-equivariant homotopy …

Remarks on the self-shrinking Clifford torus

On the one hand, we prove that the Clifford torus in $\mathbb{C}^2$ is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian $F$-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is rigid: it is locally unique as a self-shrinker for mean curvature flow, despite having infinitesimal deformations which do not arise from rigid motions. The proofs rely on analysing higher order phenomena: specifically, showing that the Clifford torus is not a local entropy minimi…

Topology of complexity one quotients

We describe of the topology of the geometric quotients of 2n dimensional compact connected symplectic manifolds with n-1 dimensional torus actions. When the isotropy weights at each fixed point are in general position, the quotient is homeomorphic to a sphere.

Quasimap Floer cohomology and varying symplectic quotients

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.

哈密顿环面作用与奇贝蒂数的单峰性

Hamiltonian torus actions and the unimodality of odd Betti numbers

研究具有非平凡哈密顿环面作用的闭辛流形奇贝蒂数序列是否单峰，通过逐渐加强对称假设，在维度6、8和10下为该问题提供肯定证据。

辛环面商簇的同构

Isomorphisms of Symplectic Torus Quotients

本文研究辛环面商簇的同构问题，证明在余维数至少为4时商簇决定原模和群，并在余维数为3时通过构造子群和子模实现约化，同时推广到实情形。

Chekanov torus and Gelfand--Zeitlin torus in $S^2 \times S^2$

The Chekanov torus was the first known \emph{exotic} torus, a monotone Lagrangian torus that is not Hamiltonian isotopic to the standard monotone Lagrangian torus. We explore the relationship between the Chekanov torus in $S^2 \times S^2$ and a monotone Lagrangian torus which had been introduced before Chekanov's construction \cite{Chekanov}. We prove that the monotone Lagrangian torus fiber in a certain Gelfand--Zeitlin system is Hamiltonian isotopic to the Chekanov torus in $S^2 \times S^2$.

The cohomology rings of abelian symplectic quotients

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.

Bulky Hamiltonian isotopies of Lagrangian tori with applications

We exhibit monotone Lagrangian tori inside the standard symplectic four-dimensional unit ball that become Hamiltonian isotopic to the Clifford torus, i.e.~the standard product torus, only when considered inside a strictly larger ball (they are not even symplectomorphic to a standard torus inside the unit ball). These tori are then used to construct new examples of symplectic embeddings of toric domains into the unit ball which are symplectically knotted in the sense of J.~Gutt and M.~Usher. We also give a characterisation of the Clifford torus inside the ball as well as the projective plane in…

Polarized orbifolds associated to quantized Hamiltonian torus actions

Suppose given an holomorphic and Hamiltonian action of a compact torus $T$ on a polarized Hodge manifold $M$. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of $T$ on the associated Hardy space. If in addition the moment map is nowhere zero, for each weight $\boldsymbolν$ the $\boldsymbolν$-th isotypical component in the Hardy space of the polarization is finite-dimensional. Assuming that the moment map is transverse to the ray through $\boldsymbolν$, we give a gometric interpretation of the isotypical components associated to the…

Desingularisation of orbifolds obtained from symplectic reduction at generic coadjoint orbits

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…

Deformation of Dirac operators along orbits and quantization of non-compact Hamiltonian torus manifolds

We give a formulation of a deformation of Dirac operator along orbits of a group action on a possibly non-compact manifold to get an equivariant index and a K-homology cycle representing the index. We apply this framework to non-compact Hamiltonian torus manifolds to define geometric quantization from the view point of index theory. We give two applications. The first one is a proof of a [Q,R]=0 type theorem, which can be regarded as a proof of the Vergne conjecture for Abelian case. The other is a Danilov-type formula for toric case in the non-compact setting, which shows that this geometric …

The K-theory of abelian symplectic quotients

Let T be a compact torus and (M,ω) a Hamiltonian T-space. In a previous paper, the authors showed that the T-equivariant K-theory of the manifold M surjects onto the ordinary integral K-theory of the symplectic quotient M \mod T of M by T, under certain technical conditions on the moment map. In this paper, we use equivariant Morse theory to give a method for computing the K-theory of the symplectic quotient by obtaining an explicit description of the kernel of the surjection κ: K^*_T(M) \onto K^*(M \mod T). Our results are K-theoretic analogues of the work of Tolman and Weitsman for Borel equ…

The biinvariant diagonal class for Hamiltonian torus actions

Suppose that an algebraic torus $G$ acts algebraically on a projective manifold $X$ with generically trivial stabilizers. Then the Zariski closure of the set of pairs $\{(x,y)\in X\times X\mid y=gx \text{for some}g\in G\}$ defines a nonzero equivariant cohomology class $[Δ_G]\in H^*_{G\times G}(X\times X)$. We give an analogue of this construction in the case where $X$ is a compact symplectic manifold endowed with a hamiltonian action of a torus, whose complexification plays the role of $G$. We also prove that the Kirwan map sends the class $[Δ_G]$ to the class of the diagonal in each symplect…

Log-concavity of complexity one Hamiltonian torus actions

Let $(M,ω)$ be a closed $2n$-dimensional symplectic manifold equipped with a Hamiltonian $T^{n-1}$-action. Then Atiyah-Guillemin-Sternberg convexity theorem implies that the image of the moment map is an $(n-1)$-dimensional convex polytope. In this paper, we show that the density function of the Duistermaat-Heckman measure is log-concave on the image of the moment map.

Non-Compact Symplectic Toric Manifolds

A key result in equivariant symplectic geometry is Delzant's classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular ("Delzant") polytope; this gives a bijection between unimodular polytopes and isomorphism classes of compact connected symplectic toric manifolds. In this paper we extend Delzant's classification to non-compact symplectic toric manifolds. For a non-compact symplectic toric manifold the image of the moment map nee…

Equivariant cohomology for Hamiltonian torus actions on symplectic orbifolds

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.

Hamiltonian unknottedness of certain monotone Lagrangian tori in $S^2\times S^2$

We prove that a monotone Lagrangian torus in $S^2\times S^2$ which suitably sits in a symplectic fibration with two sections in its complement is Hamiltonian isotopic to the Clifford torus.

Asymptotics of Szegö kernels under Hamiltonian torus actions

Let $X$ be the circle bundle associated to a positive line bundle on a complex projective (or, more generally, compact symplectic) manifold. The Tian-Zelditch expansion on $X$ may be seen as a local manifestation of the decomposition of the (generalized) Hardy space $H(X)$ into isotypes for the $S^1$-action. More generally, given a compatible action of a compact Lie group, and under general assumptions guaranteeing finite dimensionality of isotypes, we may look for asymptotic expansions locally reflecting the equivariant decomposition of $H(X)$ over the irreducible representations of the group…

A Note on Higher Cohomology Groups of Kähler Quotients

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 …

Gromov-Witten invariants of symplectic quotients and adiabatic limits

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…

Localization and Specialization for Hamiltonian Torus Actions

We consider a Hamiltonian action of n-dimensional torus, T^n, on a compact symplectic manifold (M,ω) with d isolated fixed points. For every fixed point p there exists (though not unique) a class a_p in H^*_{T}(M; Q) such that the collection {a_p}, over all fixed points, forms a basis for H^*_{T}(M; Q) as an H^*(BT; Q) module. The map induced by the inclusion, ι^*:H^*_{T}(M; Q) \rightarrow H^*_{T}(M^{T}; Q)= \oplus_{j=1}^{d}Q[x_1, ..., x_n] is injective. We use such classes {a_p} to give necessary and sufficient conditions for f=(f_1, ...,f_d) in \oplus_{j=1}^{d}Q[x_1, ..., x_n] to be in the i…

$SU(3)$上的双侧环面作用与复几何

Double sided torus actions and complex geometry on $SU(3)$

本研究在$SU(3)$上构造对应复二次曲面实二次方程变体的显式复结构与横截Kähler全纯叶状结构，推广了已有不变结构，并得到旗簇$SU(3)/T$的轨形变体。