Related papers: The fundamental group of symplectic manifolds with…

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…

We associate to each symplectic $4$-orbifold $X$ a canonical smooth symplectic resolution $\pi: \tilde{X}\rightarrow X$, which can be done equivariantly if $X$ comes with a symplectic $G$-action by a finite group. Moreover, we show that the…

We study the action of a real reductive group $G$ on a Kahler manifold $Z$ which is the restriction of a holomorphic action of a complex reductive Lie group $U^\mathbb{C}.$ We assume that the action of $U$, a maximal compact connected…

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…

Let $M=(M,\omega)$ be either the product $S^2\times S^2$ or the non-trivial $S^2$ bundle over $S^2$ endowed with any symplectic form $\omega$. Suppose a finite cyclic group $Z_n$ is acting effectively on $(M,\omega)$ through Hamiltonian…

Let $G$ be a complex semisimple Lie group and ${G}_{\mathbb R}$ a real form that contains a compact Cartan subgroup $T_{\mathbb R}$. Let $\pi$ be a discrete series representation of $G_{\mathbb R}$. We present geometric interpretations in…

In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain $\omega$ on a Lie algebra $h$ with values in an $h$-module $V$, we associate subalgebras $sp(h,\omega) \supeq…

We show that the fundamental group of every enumeratively rationally connected closed symplectic manifold is finite. In other words, if a closed symplectic manifold has a non-zero Gromov-Witten invariant with two point insertions, then it…

Let $G$ be a Lie group, $H$ a closed subgroup and $M$ the homogeneous space $G/H$. Each representation $\Psi$ of $H$ determines a $G$-equivariant principal bundle ${\mathcal P}$ on $M$ endowed with a $G$-invariant connection. We consider…

For a compact, connected, simply-connected Lie group G, the loop group LG is the infinite-dimensional Hilbert Lie group consisting of H^1-Sobolev maps S^1-->G. The geometry of LG and its homogeneous spaces is related to representation…

Let ${\cal O}$ be a quantizable coadjoint orbit of a semisimple Lie group $G$. Under certain hypotheses we prove that $#(\pi_1(\text{Ham}({\cal O})))\geq #(Z(G))$, where $\text{Ham}({\cal O})$ is the group of Hamiltonian symplectomorphisms…

This paper investigates ways to enlarge the Hamiltonian subgroup Ham of the symplectomorphism group Symp(M) of the symplectic manifold (M, \omega) to a group that both intersects every connected component of Symp(M) and characterizes…

We establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a noncompact symplectic manifold with proper moment map.

Let $\varphi$ and $\varphi'$ be two homotopic actions of the topological group $G$ on the topological space $X$. To an object $A$ in the $G$-equivariant derived category $D_{\varphi}(X)$ of $X$ relative to the action $\varphi$ we associate…

We classify compact, connected Hamiltonian and quasi-Hamiltonian manifolds of cohomogeneity one (which is the same as being multiplicity free of rank one). Here the group acting is a compact connected Lie group (simply connected in the…

Symplectic slice theorems elucidate the local structure of symplectic manifolds carrying Hamiltonian actions of compact Lie groups. We generalize these theorems in two natural settings. The first is based on the idea that complex reductive…

Let $(M, \omega)$ be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point set consists of isolated points or surfaces. Assume dim $H^2(M)<3$, in \cite{L}, we…

We consider canonical symplectic structure on the moduli space of flat ${\g}$-connections on a Riemann surface of genus $g$ with $n$ marked points. For ${\g}$ being a semisimple Lie algebra we obtain an explicit efficient formula for this…

Let $(M,\omega)$ be a compact symplectic manifold with a Hamiltonian GKM action of a compact torus. We formulate a positive condition on the space; this condition is satisfied if the underlying symplectic manifold is monotone. The main…

In this paper we study Poisson actions of complete Poisson groups, without any connectivity assumption or requiring the existence of a momentum map. For any complete Poisson group $G$ with dual $G^\star$ we obtain a suitably connected…