Related papers: Generalized complex hamiltonian torus actions: Exa…

We show that if a holomorphic $n$ dimensional compact torus action on a compact connected complex manifold of complex dimension $n$ has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.

Let $M$ be a compact symplectic manifold on which a compact torus $T$ acts Hamiltonialy with a moment map $\mu$. Suppose there exists a symplectic involution $\theta:M\to M$, such that $\mu\circ\theta=-\mu$. Assuming that 0 is a regular…

In this paper, we study the hard Lefschetz property of a symplectic manifold which admits a Hamiltonian torus action. More precisely, let $(M,\omega)$ be a 6-dimensional compact symplectic manifold with a Hamiltonian $T^2$-action. We will…

We give an affirmative answer to the Halperin-Carlsson conjecture for the homologically injective torus actions on closed manifolds. This class contains holomorphic torus actions on compact Kahler manifolds, torus actions on compact…

In this present paper we study geometry of compact complex manifolds equipped with a \emph{maximal} torus $T=(S^1)^k$ action. We give two equivalent constructions providing examples of such manifolds given a simplicial fan $\Sigma$ and a…

In this paper we first consider the Hamiltonian action of a compact connected Lie group on an $H$-twisted generalized complex manifold $M$. Given such an action, we define generalized equivariant cohomology and generalized equivariant…

We show that for a Hamiltonian action of a compact torus $G$ on a compact, connected symplectic manifold $M$, the $G$-equivariant cohomology is determined by the residual $S^1$ action on the submanifolds of $M$ fixed by codimension-1 tori.…

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…

Given a symplectic manifold, we ask in how many different ways can a torus act on it. Classification theorems in equivariant symplectic geometry can sometimes tell that two Hamiltonian torus actions are inequivalent, but often they do not…

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 $\T^n = \R^{n}/2\pi\Z^n$, is a smooth projective toric variety completely…

A Hamiltonian action of a complex torus on a symplectic complex manifold is said to be {\it multiplicity free} if a general orbit is a lagrangian submanifold. To any multiplicity free Hamiltonian action of a complex torus $T\cong…

In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds,…

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…

Let M be a Kaehler manifold with a free, holomorphic and Hamiltonian action of the standard n-torus T. We give a simple, explicit and canonical formula for the Kaehler potential on the Kaehler reduction of M. As a consequence we can derive…

A manifold obtained by k simultaneous symplectic blow-ups of CP^2 of equal sizes epsilon (where the size of CP^1 in CP^2 is one) admits an effective two-dimensional torus action if k <= 3. We show that it does not admit such an action if k…

We consider effective actions of a compact torus $T^{n-1}$ on an even-dimensional smooth manifold $M^{2n}$ with isolated fixed points. We prove that under certain conditions on weights of tangent representations, the orbit space is a…

We show that if $(X,g,J,\omega)$ is a K\"ahler manifold with an $SU(n+s)$-structure and a Hamiltonian holomorphic action of a compact torus $T^s$, then the usual symplectic quotient $Y$ inherits an $SU(n)$-structure provided the existence…

Consider a smooth effective action of a torus $\mathbb{T}^n$ on a connected $C^{\infty}$-manifold $M$ of dimension $m$. Then $n\leq m$. In this work we show that if $n<m$, then there exist a complete vector field $X$ on $M$ such that the…

Suppose given a Hamiltonian and holomorphic action of $G=U(2)$ on a compact K\"{a}hler manifold $M$, with nowhere vanishing moment map. Given an integral coadjoint orbit $\mathcal{O}$ for $G$, under transversality assumptions we shall…

In the first part of this paper, we let $G$ be a finitely-generated amenable group such that $G/[G, G]$ is torsion-free. We suppose that $G$ acts by homeomorphisms homotopic to the identity on a manifold $M$, and give conditions on $M$…