Paul-Emile Paradan
Let Z be the real part of a K{ä}hler Hamiltonian manifold M. The O'Shea-Sjamaar's Theorem tells us that the moment polytope Delta(Z) corresponds to the anti-invariant part of the Kirwan polytope Delta(M). The purpose of the present paper is to explain how to parameterize the equations of the facets of Delta(Z) in terms of real Ressayre's pairs of Z.
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In this article we consider a generalization of manifolds and orbifolds which we call quasifolds; quasifolds of dimension k are locally isomorphic to the quotient of R^k by the action of a discrete group - tipically they are not Hausdorff topological spaces. The analogue of a torus in this geometry is a quasitorus. We define Hamiltonian actions of quasitori on symplectic quasifolds and we show that any simple convex polytope, rational or not, is the image of the moment mapping for a family of effective Hamiltonian actions on symplectic quasifolds having twice the dimension of the corresponding…
In this work, we study some convex cones associated to isotropic representations of symmetric spaces. We explain the inequalities that describe them by means of cohomological conditions. In particular, we study the singular Horn cone which is the counterpart of the classical Horn cone, where the eigenvalues of Hermitian square matrices are replaced by the singular values of rectangular matrices.
Kirwan polytopes in the real setting
本文研究实数情形下的Kirwan多边形，整合并完善了之前两篇预印本的研究成果。
We apply a local differential geometric framework from Kähler toric geometry to (re)construct Calabi's extremal Kähler metrics on $\bbC\bbP^n$ blown-up at a point from data on the moment polytope.
In the classical theory of toric manifolds polytopes appear in two guises -- as Newton polytopes of line bundles on the complex, and as moment polytopes on the symplectic side, the link between the two being established by the prequantizability condition on the cohomology class of the symplectic form. Here we give a combinatorial description of the orbifold Picard group for complete toric orbifolds, with the aim of detailing the relation between complex and symplectic aspects in the orbifold setting. In particular this permits to illustrate the breakdown of identification of (orbifold) line bu…
We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We introduce the tropical momentum map, which takes values in a generalization of affine space called a log affine manifold. Using this momentum map, we obtain a complete classification of such manifolds in terms of decorated log affine polytopes, hence extending the classification of symplectic toric manifolds achieved by Atiyah, Guillemin-Sternberg, Kostant, and Delzant.
A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\T^n = \R^{n}/2π\Z^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\T^n$-space) by the image of the moment map $ϕ:M\to\R^n$, a convex polytope $P=ϕ(M)\subset\R^n$. In this paper we show, using symplectic (action-angle) coordinates on $P\times \T^n$, how all $\om$-compatible toric complex structures on $M$ can be effectively parametrized by smooth functions on $P$. We also discuss some topics suited for a…
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].
We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.
Due to a previous result which states that contact varieties are isomorphic to certain varieties, the momentum polytopes of contact manifolds are convex.
In this paper we completely classify symplectic actions of a torus $T$ on a compact connected symplectic manifold $(M, σ)$ when some, hence every, principal orbit is a coisotropic submanifold of $(M, σ)$. That is, we construct an explicit model, defined in terms of certain invariants, of the manifold, the torus action and the symplectic form. The invariants are invariants of the topology of the manifold, of the torus action, or of the symplectic form. In order to deal with symplectic actions which are not Hamiltonian, we develop new techniques, extending the theory of Atiyah, Guillemin-Sternbe…
We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of a Kähler-Ricci soliton on any toric monotone manifold on any compact convex simple labelled polytope satisfying the combinatoric condition corresponding to monotonicity. We obtain that any compact convex simple polytope $P\subset \bR^n$ admits a set of inward normals, unique up to dilatation, such that there exists a symplectic potential satisfying the Guillemin boundary condi…
Delzant type theorem for torus-equivariantly embedded toric hypersurfaces
本文研究辛环面流形中余维一复子环面轨道闭包的非奇异性条件，以多面体语言刻画，并推广了Delzant定理。
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 describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left canonical action of the group on itself is Hamiltonian. The principal tool used for our description is a canonical affine structure associated with the symplectic form. We also characterize the Hamiltonian symplectic Lie groups among the connected symplectic Lie groups. We special…
Symplectic maps and hyperKähler moment map geometry
本文建立了4维实环面辛微分同胚群与超Kähler动量映射零点轨迹的对应关系，提出修正动量映射流，推广至多面体设定并证明Duistermaat型定理，为多面体辛几何与Morse-Bott理论奠定基础。
Moment polytopes of toric exponential families
本文证明有限样本空间上指数族环化得到的Kähler环面流形的矩多面体是高维单形的投影。
Bohr-Sommerfeld quantization of $b$-symplectic toric manifolds
本文研究 $b$-辛环面流形的量子化，通过 $T$-模定义 Bohr-Sommerfeld 量子化，证明其与 [GMW18b] 的形式几何量子化一致，且维数等于矩多面体中整点的带符号计数。
Let M^{2n} be a symplectic toric manifold with a fixed T^n-action and with a toric Kähler metric g. Abreu asked whether the spectrum of the Laplace operator $Δ_g$ on $\mathcal{C}^\infty(M)$ determines the moment polytope of M, and hence by Delzant's theorem determines M up to symplectomorphism. We report on some progress made on an equivariant version of this conjecture. If the moment polygon of M^4 is generic and does not have too many pairs of parallel sides, the so-called equivariant spectrum of M and the spectrum of its associated real manifold M_R determine its polygon, up to translation …
The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphisms on the space of compatible almost complex structures that motivates study of extremal Kähler metrics. In particular, moment constant connections are critical, where a symplectic connection is \textit{critical} if it is critical, with respect to arbitrary variations, for the $L^…
Constructibility of momentum maps and linear variation for singular symplectic reduced spaces
本文证明紧群真动量映射横截像的积分仿射分层与局部平凡纤维化，据此把 Duistermaat–Heckman 线性变化定理推广至奇异约化值，并扩展到真拟辛群胚作用。
We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kähler cone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assump…
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…
We deform the moment map picture on the space of symplectic connections on a symplectic manifold. To do that, we study a vector bundle of Fedosov star product algebras on the space of symplectic connections. We describe a natural formal connection on this bundle adapted to the star product algebras on the fibers. We study its curvature and show the star product trace of the curvature is a formal symplectic form on the space of symplectic connections. The action of Hamiltonian diffeomorphisms on symplectic connections preserves the formal symplectic structure and we show the star product trace …
Anti-diagonal toric generalized K$\ddot{a}$hler structures of symplectic type on a compact toric symplectic manifold were investigated in \cite{Wang2} . In this article, we consider \emph{general} toric generalized K$\ddot{a}$hler structures of symplectic type, without requiring them to be anti-diagonal. Such a structure is characterized by a triple $(τ, C, F)$ where $τ$ is a strictly convex function defined in the interior of the moment polytope $Δ$ and $C, F$ are two constant anti-symmetric matrices. We prove that underlying each such a structure is a \emph{canonical} toric K$\ddot{a}$hler s…
In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each facet and that all such orbifolds are algebraic toric varieties.
We study deformations of symplectic structures on a smooth manifold $M$ via the quasi-Poisson theory. By a fact, we can deform a given symplectic structure $ω$ to a new symplectic structure $ω_t$ parametrized by some element $t$ in $Λ^2\mathfrak{g}$, where $\mathfrak{g}$ is the Lie algebra of a Lie group $G$. Moreover, we can get a lot of concrete examples for the deformations of symplectic structures on the complex projective space and the complex Grassmannian.
On uniqueness of the equivariant smooth structure on a real moment-angle manifold
本文利用微积分论证推导证明：组合等价简单多面体的实力矩-角流形是等变微分同胚的，且多面体作为带角流形也是微分同胚的。
We consider a connected symplectic manifold $M$ acted on by a connected Lie group $G$ in a Hamiltonian fashion. If $G$ is compact, we prove give an Equivalence Theorem for the symplectic manifolds whose squared moment map $\parallel μ\parallel^2$ is constant. This result works also in the almost-Kähler setting. Then we study the case when $G$ is a non compact Lie group acting properly on $M$ and we prove a splitting results for symplectic manifolds.
We show that the complex projective space has maximal degree (volume) among all n-dimensional Kahler-Einstein Fano manifolds admitting a holomorphic C^*-action with a finite number of fixed points. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart's volume conjecture for a large class of rational polytopes, including duals of lattice polytopes. The case of spherical varieties/multiplicity free symplectic manifolds is also discussed. The proof uses Moser-Trudinger type inequalities for Stein domains and also leads to criticality results for mean…
Consider a Hamiltonian action of a compact Lie group on a compact symplectic manifold. A theorem of Kirwan's says that the image of the momentum mapping intersects the positive Weyl chamber in a convex polytope. I present a new proof of Kirwan's theorem, which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold. It also applies to some interesting classes of noncompact or singular Hamiltonian spaces, such as cotangent bundles and complex affine varieties.
The moment-angle complex Z_K is cell complex with a torus action constructed from a finite simplicial complex K. When this construction is applied to a triangulated sphere K or, in particular, to the boundary of a simplicial polytope, the result is a manifold. Moment-angle manifolds and complexes are central objects in toric topology, and currently are gaining much interest in homotopy theory, complex and symplectic geometry. The geometric aspects of the theory of moment-angle complexes are the main theme of this survey. We review constructions of non-Kahler complex-analytic structures on mome…
We pursue the symplectic description of toric Kahler manifolds. There exists a general local classification of metrics on toric Kahler manifolds equipped with Hamiltonian two-forms due to Apostolov, Calderbank and Gauduchon(ACG). We derive the symplectic potential for these metrics. Using a method due to Abreu, we relate the symplectic potential to the canonical potential written by Guillemin. This enables us to recover the moment polytope associated with metrics and we thus obtain global information about the metric. We illustrate these general considerations by focusing on six-dimensional Ri…
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