Related papers: Kahler geometry of toric manifolds in symplectic c…

In this article, we provide an exposition about symplectic toric manifolds, which are symplectic manifolds $(M^{2n}, \omega)$ equipped with an effective Hamiltonian $\mathbb{T}^n\cong (S^1)^n$-action. We summarize the construction of $M$ as…

We study Hamiltonian actions on $b$-symplectic manifolds with a focus on the effective case of half the dimension of the manifold. In particular, we prove a Delzant-type theorem that classifies these manifolds using polytopes that reside in…

This is a sequel of \cite{Wang}, which provides a general formalism for this paper. We mainly investigate thoroughly a subclass of toric generalized K$\ddot{a}$hler manifolds of symplectic type introduced by Boulanger in \cite{Bou}. We find…

In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an…

Suppose that a compact $r$-dimensional torus $T^r$ acts in a holomorphic and Hamiltonian manner on polarized complex $d$-dimensional projective manifold $M$, with nowhere vanishing moment map $\Phi$. Assuming that $\Phi$ is transverse to…

In this paper we completely classify symplectic actions of a torus $T$ on a compact connected symplectic manifold $(M, \sigma)$ when some, hence every, principal orbit is a coisotropic submanifold of $(M, \sigma)$. That is, we construct an…

Toric varieties play an important role both in symplectic and complex geometry. In symplectic geometry, the construction of a symplectic toric manifold from a smooth polytope is due to Delzant. In algebraic geometry, there is a more general…

A symplectic toric orbifold is a compact connected orbifold $M$, a symplectic form $\omega$ on $M$, and an effective Hamiltonian action of a torus $T$ on $M$, where the dimension of $T$ is half the dimension of $M$. We prove that there is a…

Given a compact symplectic toric manifold $(M,\omega, \mathbb{T})$, we identify a class $DGK_{\omega}^{\mathbb{T}}(M)$ of $\mathbb{T}$-invariant generalized K\"ahler structures for which a generalisation the Abreu-Guillemin theory of toric…

We give an intrinsic definition of toric symplectic stacks, and show that they are classified by simple convex polytopes equipped with some additional combinatorial data. This generalizes Delzant's classification of toric symplectic…

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…

We discuss the construction of toric Kaehler metrics on symplectic 2n-manifolds with a hamiltonian n-torus action and present a simple derivation of the Guillemin formula for a distinguished Kaehler metric on any such manifold. The results…

Let $P$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $X_{P}$ in toric K\"ahler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated…

We extend the Abreu-Guillemin theory of invariant K\"ahler metrics from toric symplectic manifolds to any symplectic manifold admitting a toric action of a symplectic torus bundle. We show that these are precisely the symplectic manifolds…

In \cite{btoric}, Guillemin et al. proved a Delzant-type theorem which classifies $b$-symplectic toric manifolds. More generally, in \cite{torus} they proved a similar convexity result for general Hamiltonian torus action on $b$-symplectic…

Delzant's theorem for symplectic toric manifolds says that there is a one-to-one correspondence between certain convex polytopes in $\mathbb{R}^n$ and symplectic toric $2n$-manifolds, realized by the image of the moment map. I review proofs…

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…

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,…

A symplectic cut of a manifold M with a Hamiltonian circle action is a symplectic quotient of M x C. If M is Kaehler then, since C is Kaehler, the cut space is Kaehler as well. The symplectic structure on the cut is well understood. In this…

We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a…