Moment maps, symplectomorphism groups and compatible complex structures

有理直纹面上 Hamilton 圆周作用的中心化子

Centralizers of Hamiltonian circle actions on rational ruled surfaces

本文研究两类有理直纹面上 Hamilton $S^1$ 作用的等变辛微分同胚群，通过近复结构分层及 Delzant、Karshon 分类确定其同伦型为环面或两环面的同伦推出。

Remarks on the naturality of quantization

Hamiltonian quantization of an integral compact symplectic manifold M depends on a choice of compatible almost complex structure J. For open sets U in the set of compatible almost complex structures and small enough values of Planck's constant, the Hilbert spaces of the quantization form a bundle over U with a natural connection. In this paper we examine the dependence of the Hilbert spaces on the choice of J, by computing the semi-classical limit of the curvature of this connection. We also show that parallel transport provides a link between the action of the group Symp(M) of symplectomorphi…

辛向量空间的格拉斯曼流形的莫尔斯-博特统一

A Morse-Bott unification of the Grassmannians of a symplectic vector space

本文提出一种基于相容复结构的二次莫尔斯-博特函数，统一了辛向量空间的各类格拉斯曼流形，并通过负梯度流将其退化为单位群的紧致齐性空间。

Contact Symplectic Fibrations and Fiber Connected Sum

We consider certain type of fiber bundles with odd dimensional compact contact base, exact symplectic fibers, and the structure group contained in the group of exact symplectomorphisms of the fiber. We call such fibrations "contact symplectic fibrations". By a result of Hajduk-Walczak, some of these admit contact structures which are "compatible" (in a certain sense) with the corresponding fibration structures. We show that this result can be extended to get a compatible contact structure on any contact symplectic fibration, and also that isotopic contact structures on the base produce isotopi…

Geometric Structures on Spaces of Weighted Submanifolds

In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold $(M,ω)$, we construct a weak symplectic structure on each leaf ${\textbf I}_{w}$ of a foliation of the space of compact oriented isotropic submanifolds in $M$ equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weig…

Almost Complex Structures on $S^2\times S^2$

In this note we investigate the structure of the space $\Jj$ of smooth almost complex structures on $S^2\times S^2$ that are compatible with some symplectic form. This space has a natural stratification that changes as the cohomology class of the form changes and whose properties are very closely connected to the topology of the group of symplectomorphisms of $S^2\times S^2$. By globalizing standard gluing constructions in the theory of stable maps, we show that the strata of $\Jj$ are Fréchet manifolds of finite codimension, and that the normal link of each stratum is a finite dimensional str…

（预）多重辛流形上李代数胚的相容 E-微分形式

Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds

本文在（预）多重辛流形上的李代数胚中引入相容 E-n-形式，统一刻画扭曲泊松结构、动量映射及其同伦推广等结构。

The Action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures

We extend the definition of Weinstein's Action homomorphism to Hamiltonian actions with equivariant moment maps of (possibly infinite-dimensional) Lie groups on symplectic manifolds, and show that under conditions including a uniform bound on the symplectic areas of geodesic triangles the resulting homomorphism extends to a quasimorphism on the universal cover of the group. We apply these principles to finite dimensional Hermitian Lie groups like Sp(2n,R), reinterpreting the Guichardet-Wigner quasimorphisms, and to the infinite dimensional groups of Hamiltonian diffeomorphisms Ham(M,\om) of cl…

Geometric structures on nilpotent Lie groups: on their classification and a distinguished compatible metric

Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the invariant Ricci flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish geome…

矩拉格朗日对应在体形变后是无障碍的

Moment Lagrangian correspondences are unobstructed after bulk deformation

该研究在假设伪全纯圆盘模空间存在特定等变Kuranishi结构与相容等变CF-扰动的条件下，证明了矩映射自由零水平集上辛约化映射诱导的拉格朗日对应经体形变后无障碍，推进了辛几何相关理论研究。

A canonical compatible metric for geometric structures on nilmanifolds

Let (N,g) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their `almost' versions). We define a left invariant Riemannian metric on N compatible with g to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the `invariant Ricci' flow and are characterized as the critical points of a natural variational problem. The uniqueness…