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Register a new userRegular Poisson manifolds of compact types (PMCT 2)
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This is the second paper of a series dedicated to the study of Poisson structures of compact types (PMCTs). In this paper, we focus on regular PMCTs, exhibiting a rich transverse geometry. We show that their leaf spaces are integral affine orbifolds. We prove that the cohomology class of the leafwise symplectic form varies linearly and that there is a distinguished polynomial function describing the leafwise sympletic volume. The leaf space of a PMCT carries a natural Duistermaat-Heckman measure and a Weyl type integration formula holds. We introduce the notion of a symplectic gerbe, and we show that they obstruct realizing PMCTs as the base of a symplectic complete isotropic fibration (a.k.a. a non-commutative integrable system).
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This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties of PMCTs, which already show that they are the analogues of compact symplectic manifolds, thus placing them in a prominent position among all Poisson manifolds. For instance, their Poisson cohomology behaves very much like the de Rham cohomology of compact symplectic manifolds (Hodge decomposition, non-degenerate Poincare duality pairing, etc.) and the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and Symplectic Topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a nontrivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian $G$-spaces, foliation theory, Lie theory and symplectic gerbes.
For a Poisson manifold $M$ we develop systematic methods to compute its Picard group $Pic(M)$, i.e., its group of self Morita equivalences. We establish a precise relationship between $Pic(M)$ and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of the identity coincide; this allows us to introduce the Picard Lie algebra of $M$ and to study its basic properties. Our methods lead, in particular, to the proof of a conjecture from [BW04] stating that for any compact simple Lie algebra $mathfrak{g}$ the group $Pic(mathfrak{g}^*)$ concides with the group of outer automorphisms of $mathfrak{g}$.
We study holomorphic GL(2) and SL(2) geometries on compact complex manifolds. We show that a compact Kahler manifold of complex even dimension higher than two admitting a holomorphic GL(2)-geometry is covered by a compact complex torus. We classify compact Kahler-Einstein manifolds and Fano manifolds bearing holomorphic GL(2)-geometries. Among the compact Kahler-Einstein manifolds we prove that the only examples bearing holomorphic GL(2)-geometry are those covered by compact complex tori, the three dimensional quadric and those covered by the three dimensional Lie ball (the non compact dual of the quadric).
We consider a class of compact homogeneous CR manifolds, that we call $mathfrak n$-reductive, which includes the orbits of minimal dimension of a compact Lie group $K_0$ in an algebraic homogeneous variety of its complexification $K$. For these manifolds we define canonical equivariant fibrations onto complex flag manifolds. The simplest example is the Hopf fibration $S^3tomathbb{CP}^1$. In general these fibrations are not $CR$ submersions, however they satisfy a weaker condition that we introduce here, namely they are CR-deployments.
A {em 2-Riemannian manifold} is a differentiable manifold exhibiting a 2-inner product on each tangent space. We first study lower dimensional 2-Riemannian manifolds by giving necessary and sufficient conditions for flatness. Afterward we associate to each 2-Riemannian manifold a unique torsion free compatible pseudoconnection. Using it we define a curvature for 2-Riemannian manifolds and study its properties. We also prove that 2-Riemannian pseudoconnections do not have Koszul derivatives. Moreover, we define stationary vector field with respect to a 2-Riemannian metric and prove that the stationary vector fields in $mathbb{R}^2$ with respect to the 2-Riemannian metric induced by the Euclidean product are the divergence free ones.
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