Reduction,Induction and Ricci flat symplectic connections

A moment map interpretation of the Ricci form, Kähler--Einstein structures, and Teichmüller spaces

This paper surveys the role of moment maps in Kähler geometry. The first section discusses the Ricci form as a moment map and then moves on to moment map interpretations of the Kähler--Einstein condition and the scalar curvature (Quillen--Fujiki--Donaldson). The second section examines the ramifications of these results for various Teichmüller spaces and their Weil--Petersson symplectic forms and explains how these arise naturally from the construction of symplectic quotients. The third section discusses a symplectic form introduced by Donaldson on the space of Fano complex structures.

Homogeneous symplectic manifolds with Ricci-type curvature

We consider invariant symplectic connections $\nabla$ on homogeneous symplectic manifolds $(M,ω)$ with curvature of Ricci type. Such connections are solutions of a variational problem studied by Bourgeois and Cahen, and provide an integrable almost complex structure on the bundle of almost complex structures compatible with the symplectic structure. If $M$ is compact with finite fundamental group then $(M,ω)$ is symplectomorphic to $¶_n(\C)$ with a multiple of its Kähler form and $\nabla$ is affinely equivalent to the Levi-Civita connection.

Symplectic Connections of Ricci Type and Star Products

In this article we relate the construction of Ricci type symplectic connections by reduction to the construction of star product by reduction yielding rather explicit descriptions for the star product on the reduced space.

Moduli space of symplectic connections of Ricci type on $T^{2n}$; a formal approach

We consider analytic curves $\nabla^t$ of symplectic connections of Ricci type on the torus $T^{2n}$ with $\nabla^0$ the standard connection. We show, by a recursion argument, that if $\nabla^t$ is a formal curve of such connections then there exists a formal curve of symplectomorphisms $ψ_t$ such that $ψ_t\cdot\nabla^t$ is a formal curve of flat invariant symplectic connections and so $\nabla^t$ is flat for all $t$. Applying this result to the Taylor series of the analytic curve, it means that analytic curves of symplectic connections of Ricci type starting at $\nabla^0$ are also flat. The gr…

Minimal coadjoint orbits and symplectic induction

Let $(X,ω)$ be an integral symplectic manifold and let $(L,\nabla)$ be a quantum line bundle, with connection, over $X$ having $ω$ as curvature. With this data one can define an induced symplectic manifold $(\widetilde {X},ω_{\widetilde {X}})$ where $dim \widetilde {X} = 2 + dim X$. It is then shown that prequantization on $X$ becomes classical Poisson bracket on $\widetilde {X}$. We consider the possibility that if $X$ is the coadjoint orbit of a Lie group $K$ then $\widetilde {X}$ is the coadjoint orbit of some larger Lie group $G$. We show that this is the case if $G$ is a non-compact simpl…

Non-formal deformation quantizations of solvable Ricci-type symplectic symmetric spaces

Ricci-type symplectic manifolds have been introduced and extensively studied by M. Cahen et al.. In this note, we describe their deformation quantizations in the split solvable symmetric case. In particular, we introduce the notion of non-formal tempered deformation quantization on such a space. We show that the set of tempered deformation quantizations is in one-to-one correspondence with the space of Schwartz operator multipliers on the real line. Moreover we prove that every invariant formal star product on a split Ricci-type solvable symmetric space is an asymptotic expansion of a tempered…

Symplectic connections

This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far reaching generalization to special connections. A twistorial construction shows a relation between Ricci-type connections and complex geometry. We give a construction of Ricci-flat symplectic connections. We end up by presenting, through an explicit example,…

Induction of Hamiltonian Poisson actions

We propose a Poisson-Lie analog of the symplectic induction procedure, using an appropriate Poisson generalization of the reduction of symplectic manifolds with symmetry. Having as basic tools the equivariant momentum maps of Poisson actions, the double group of a Poisson-Lie group and the reduction of Poisson manifolds with symmetry, we show how one can induce a Poisson action admitting an equivariant momentum map. We prove that, under certain conditions, the dressing orbits of a Poisson-Lie group can be obtained by Poisson induction from the dressing orbits of a Poisson-Lie subgroup.

Global symplectic coordinates on gradient Kaehler-Ricci solitons

A classical result of D. McDuff asserts that a simply-connected complete Kaehler manifold $(M,g,ω)$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $Ψ: M\rightarrow R^{2n}$ (where $n$ is the complex dimension of $M$), satisfying the following property (proved by E. Ciriza): the image $Ψ(T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $Ψ(p)=0$, is a complex linear subspace of $C^n \simeq R^{2n}$. The aim of this paper is to exhibit, for all positive integers $n$, examples of $n$-dimensional compl…

Homogeneous and locally homogeneous solutions to symplectic curvature flow

J. Streets and G. Tian recently introduced symplectic curvature flow, a geometric flow on almost Kähler manifolds generalising Kähler-Ricci flow. The present article gives examples of explicit solutions to this flow of non-Kähler structures on several nilmanifolds and on twistor fibrations over hyperbolic space studied by J. Fine and D. Panov. The latter lead to examples of non-Kähler static solutions of symplectic curvature flow which can be seen as analogues of Kähler-Einstein manifolds in Kähler-Ricci flow.

Special homogeneous almost complex structures on symplectic manifolds

Homogeneous compatible almost complex structures on symplectic manifolds are studied, focusing on those which are special, meaning that their Chern-Ricci form is a multiple of the symplectic form. Non Chern-Ricci flat ones are proven to be covered by co-adjoint orbits. Conversely, compact isotropy co-adjoint orbits of semi-simple Lie groups are shown to admit special compatible almost complex structures whenever they satisfy a necessary topological condition. Some classes of examples including twistor spaces of hyperbolic manifolds and discrete quotients of Griffiths period domains of weight t…

Symplectic maps to projective spaces and symplectic invariants

After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of CP^2, we describe a new construction of maps from symplectic manifolds of any dimension to CP^2 and the associated monodromy invariants. We also show that a dimensional induction process makes it possible to describe any compact symplectic manifold by a series of words in braid groups and a word in a symmetric group.

A construction of symplectic connections through reduction

We give an elementary construction of symplectic connections through reduction. This provides an elegant description of a class of symmetric spaces and gives examples of symplectic connections with Ricci type curvature, which are not locally symmetric; the existence of such symplectic connections was unknown.

带有整体Grauert管的李群上的伪-凯勒诱导

Pseudo--Kähler induction on Lie groups with entire Grauert tubes

本文研究带有相容复结构的哈密顿$H$-空间$Y$的诱导构造，提出将标准诱导中$T^*G$替换为$G$的Grauert管的伪-凯勒诱导方法，聚焦于容许整体Grauert管的李群开展研究。

Some pseudo-Kähler Einstein $4$-symmetric spaces with a "twin" special almost complex structure

On $4$-symmetric symplectic spaces, invariant almost complex structures -- up to sign -- arise in pairs. We exhibit some $4$-symmetric symplectic spaces, with a pair of "natural" compatible (usually not positive) invariant almost complex structures, one of them being integrable and the other one being maximally non integrable (i.e. the image of its Nijenhuis tensor at any point is the whole tangent space at that point). The integrable one defines a pseudo-Kähler Einstein metric on the manifold, and the non integrable one is Ricci Hermitian (in the sense that the almost complex structure preser…

Homogeneous symplectic manifolds of Poisson-Lie groups

Symplectic manifolds which are homogeneous spaces of Poisson-Lie groups are studied in this paper. We show that these spaces are, under certain assumptions, covering spaces of dressing orbits of the Poisson-Lie groups which act on them. The effect of the Poisson induction procedure on such spaces is also examined, thus leading to an interesting generalization of the notion of homogeneous space. Some examples of homogeneous spaces of Poisson-Lie groups are discussed in the light of the previous results.

Symmetries of the Space of Linear Symplectic Connections

There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen-Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson st…

Complex of twistor operators in symplectic spin geometry

For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic analogues of the twistor operators known from Riemannian spin geometry. We prove that under the condition the symplectic Weyl curvature tensor field of the symplectic connection vanishes, the mentioned sequence forms a complex. This gives rise to a new complex for the so called Ricc…

Symplectic geometric flows

Several geometric flows on symplectic manifolds are introduced which are potentially of interest in symplectic geometry and topology. They are motivated by the Type IIA flow and T-duality between flows in symplectic geometry and flows in complex geometry. Examples include the Hitchin gradient flow on symplectic manifolds, and a new flow which is called the dual Ricci flow.

On the ellipticity of symplectic twistor complexes

For a Fedosov manifold (symplectic manifold equipped with a symplectic torsion-free affine connection $\nabla$) admitting a metaplectic structure, we shall investigate two sequences of first order differential operators acting on sections of certain bundles over this manifold. The operators are symplectic analogues of the twistor operators known from Riemannian spin geometry. Therefore we call the mentioned sequences symplectic twistor sequences. These sequences are complexes if the connection $\nabla$ is of Ricci type. We shall prove that the so called truncated parts of these complexes are e…

A locally conformally symplectic structure on Kerr space-time

R Kerr's Ricci-flat Lorentz 4-manifold was shown by B Carter in 1968 to support a classically completely integrable system of geodesics; here we interpret this system in terms of a locally conformally symplectic structure, which we identify (§2.2) using characteristic classes defined by Kerr-Schild Cartesian coordinates (regarded as a map to the Cayley-Penrose compactification of Minkowski space). This leads to the definition of an interesting cobordism category of contact 3-manifolds and 4-dimensional locally conformally symplectic cobordisms between them.

Almost Kaehler geometry of adjoint orbits of semisimple Lie groups

We study the almost Kaehler geometry of adjoint orbits of non-compact real semisimple Lie groups endowed with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almost complex structure. We give explicit formulas for the Chern-Ricci form, the Hermitian scalar curvature and the Nijenhuis tensor in terms of root data. We also discuss when the Chern-Ricci form is a multiple of the symplectic form, and when compact quotients of these orbits are of Kaehler type.

Curvature of scalar-flat Kahler metrics on non-compact symplectic toric 4-manifolds

In this paper, we show that the complete scalar-flat Kahler metrics constructed by Abreu and the author on strictly unbounded toric 4-dimensional orbifolds have finite $L^2$ norm of the full Riemannian tensor. In particular, this answers a question of Donaldon's on the corresponding Generalized Taub-NUT metric on $R^4$. This norm is explicitly determined when the underlying toric manifold is the minimal resolution of a cyclic singularity. In the Ricci-flat case corresponding to gravitational instantons, this recovers a recent result of Atiyah-Lebrun.

Moduli spaces of Einstein-Hermitian generalized connections over generalized Kahler manifolds of symplectic type

From a view point of the moment map, we shall introduce the notion of Einstein-Hermitian generalized connections over a generalized Kähler manifold of symplectic type. We show that moduli spaces of Einstein-Hermitian generalized connections arise as the Kähler quotients. The deformation complex of Einstein-Hermitian generalized connections is an elliptic complex and it turns out that the smooth part of the moduli space is a finite dimensional Kähler manifold. The canonical line bundle over a generalized Kähler manifold of symplectic type has the canonical generalized connection and its curvatu…

辛约化齐性空间上的一些不变联络

Some invariant connections on symplectic reductive homogeneous spaces

本文研究辛约化齐性空间上的自然辛联络，引入一族不变联络并证明存在唯一自然辛联络，计算其曲率张量，并验证Wallach旗流形的首选联络与之重合且Ricci平行。

Structure of the curvature tensor on symplectic spinors

We study symplectic manifolds $(M^{2l},ω)$ equipped with a symplectic torsion-free affine (also called Fedosov) connection $\nabla$ and admitting a metaplectic structure. Let $\mathcal{S}$ be the so called symplectic spinor bundle and let $R^S$ be the curvature tensor field of the symplectic spinor covariant derivative $\nabla^S$ associated to the Fedosov connection $\nabla.$ It is known that the space of symplectic spinor valued exterior differential 2-forms, $Γ(M,\bigwedge^2T^*M\otimes {\mathcal{S}}),$ decomposes into three invariant spaces with respect to the structure group, which is the m…

超凯勒四维平坦环面上辛梯度流的局部存在性

Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus

通过引入矩映射并利用DeTurck技巧将辛梯度流转化为严格抛物型流，证明了实四维环面辛同胚群的局部可缩性，方法基于Yann Rollin的思想。

Scalar-flat Kahler metrics on non-compact symplectic toric 4-manifolds

In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In this paper we generalize Donaldson's method and construct complete scalar-flat toric Kahler metrics on any symplectic toric 4-manifold with "strictly unbounded" moment polygon. These include the asymptotic locally Euclidean scalar-flat Kahler metrics previously constructed by Ca…

Symplectic 4-manifolds via Lorentzian geometry

We observe that, in dimension four, symplectic forms may be obtained via Lorentzian geometry; in particular, null vector fields can give rise to exact symplectic forms. That a null vector field is nowhere vanishing yet orthogonal to itself is essential to this construction. Specifically, we show that on a Lorentzian 4-manifold $(M,g)$, if ${\boldsymbol k}$ is a complete null vector field with geodesic flow along which $\text{Ric}({\boldsymbol k},{\boldsymbol k}) > 0$, and if $f$ is any smooth function on $M$ with ${\boldsymbol k}(f)$ nowhere vanishing, then $dg(e^f{\boldsymbol k},\cdot)$ is a …

Picard-Lefschetz theory and dilating C^*-actions

We consider C^*-actions on Fukaya categories of exact symplectic manifolds. Such actions can be constructed by dimensional induction, going from the fibre of a Lefschetz fibration to its total space. We explore applications to the topology of Lagrangian submanifolds, with an emphasis on ease of computation.

Homogeneous symplectic 4-manifolds and finite dimensional Lie algebras of symplectic vector fields on the symplectic 4-space

We classify the finite type (in the sense of E. Cartan theory of prolongations) subalgebras $\mathfrak{h}\subset\mathfrak{sp}(V)$, where $V$ is the symplectic 4-dimensional space, and show that they satisfy $\mathfrak{h}^{(k)}=0$ for all $k>0$. Using this result, we reduce the problem of classification of graded transitive finite-dimensional Lie algebras $\mathfrak{g}$ of symplectic vector fields on $V$ to the description of graded transitive finite-dimensional subalgebras of the full prolongations $\mathfrak{p}_1^{(\infty)}$ and $\mathfrak{p}_2^{(\infty)}$, where $\mathfrak{p}_1$ and $\mathfr…

Limits of Riemannian 4-manifolds and the symplectic geometry of their twistor spaces

The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $ω$. This article studies limits of manifolds for which $ω$ tames either $J_+$ or $J_-$. This amounts to a curvature inequality involving self-dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti-self-dual Einstein manifolds with non-zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperkähler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorph…

Infinite dimensional moment map geometry and closed Fedosov's star products

We study the Cahen-Gutt moment map on the space of symplectic connections of a symplectic manifold. On a Kähler manifold, we define a Calabi-type functional $\mathscr{F}$ on the space of Kähler metrics in a given Kähler class. We study the zeroes of $\mathscr{F}$. Given a zero of $\mathscr{F}$ with non-negative Ricci tensor, we show the space of zeroes around the given one has the structure of a finite dimensional embedded submanifold. We give a new motivation, coming from deformation quantisation, for the study of moment maps on infinite dimensional spaces. More precisely, we establish a stro…