Related papers: Strong Lefschetz property under reduction

We study the hard Lefschetz property on compact symplectic solvmanifolds, i.e., compact quotients $M=\Gamma\backslash G$ of a simply-connected solvable Lie group $G$ by a lattice $\Gamma$, admitting a symplectic structure.

In this paper we construct six-dimensional compact non-K\"ahler Hamiltonian circle manifolds which satisfy the strong Lefschetz property themselves but nevertheless have a non-Lefschetz symplectic quotient. This provides the first known…

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…

In this paper, we give a new method to construct a compact symplectic manifold which does not satisfy the hard Lefschetz property. Using our method, we construct a simply connected compact K\"ahler manifold $(M,J,\omega)$ and a symplectic…

When a complex semisimple group $G$ acts holomorphically on a K\"ahler manifold $(X,\omega)$ such that a maximal compact subgroup $K\subset G$ preserves the symplectic form $\omega$, a basic result of symplectic geometry says that the…

We provide new families of compact complex manifolds with no K\"ahler structure carrying symplectic structures satisfying the \textit{Hard Lefschetz Condition}. These examples are obtained as compact quotients of the solvable Lie group…

In this paper we give an explicit construction of a symplectic Lefschetz fibration whose total space is a smooth compact four dimensional manifold with a prescribed fundamental group. We also study the numerical properties of the sections…

For a Lie group $G=\R^{n}\ltimes_{\phi}\R^{m}$ with the semi-simple action $\phi:\R^{n}\to {\rm Aut}(\R^{m})$, we show that if $\Gamma$ is a finite extension of a lattice of $G$ then $K(\Gamma, 1)$ is formal. Moreover we show that a compact…

Let $(M, \omega)$ be a connected, compact symplectic manifold equipped with a Hamiltonian $G$ action, where $G$ is a connected compact Lie group. Let $\phi$ be the moment map. In \cite{L}, we proved the following result for $G=S^1$ action:…

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex…

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is…

We study restrictions on cohomology algebras of Kaehler compact manifolds, not depending on the h^{p,q} numbers or the symplectic structure. To illustrate the effectiveness of these restrictions, we give a number of examples of compact…

For Lie groups $G$ of the form $G = \R^k \ltimes_{\phi} \R^m$, with $k + m$ even, a result of H. Kasuya shows that if the action $\phi:\R^k \to \mathrm{Aut}(\R^m)$ is semisimple then any symplectic solvmanifold $(\Gamma \backslash G,…

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…

Given a symplectic manifold $(M,\omega)$ endowed with a proper Hamiltonian action of a Lie group $G$, we consider the action induced by a Lie subgroup $H$ of $G$. We propose a construction for two compatible Witt-Artin decompositions of the…

The special structures that arise in symplectic topology (particularly Gromov--Witten invariants and quantum homology) place as yet rather poorly understood restrictions on the topological properties of symplectomorphism groups. This…

We consider 6-manifolds endowed with a symplectic half-flat SU(3)-structure and acted on by a transitive Lie group G of automorphisms. We review a classical result allowing to show the non-existence of compact non-flat examples. In the…

Let $(M, \omega)$ be a connected compact symplectic manifold equipped with a Hamiltonian SU(2) or SO(3) action. We prove that, as fundamental group of topological spaces, $\pi_1(M)=\pi_1(M_{red})$, where $M_{red}$ is the symplectic quotient…

We prove several versions of "quantization commutes with reduction" for circle actions on manifolds that are not symplectic. Instead, these manifolds possess a weaker structure, such as a spin^c structure. Our theorems work whenever the…

In this note we prove the following theorem: Let $G$ be a compact Lie group acting on a compact symplectic manifold $M$ in a Hamiltonian fashion. If $L$ is an $l$-dimensional closed invariant submanifold of $M$, on which the $G$-action is…