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Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds

Laurent Meersseman (2011)

Annales scientifiques de l'École Normale Supérieure

Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of 0 in ℂ p , for some p > 0 ) or differentiable (parametrized by an open neighborhood of 0 in ℝ p , for some p > 0 ) deformation families of compact complex manifolds. Assume they are pointwise isomorphic, that is for each point t of the parameter space, the fiber over t of the first family is biholomorphic to the fiber over t of the second family. Then, under which conditions are the...

Foliations and spinnable structures on manifolds

Itiro Tamura (1973)

Annales de l'institut Fourier

In this paper we study a new structure, called a spinnable structure, on a differentiable manifold. Roughly speaking, a differentiable manifold is spinnable if it can spin around a codimension 2 submanifold, called the axis, as if the top spins.The main result is the following: let M be a compact ( n - 1 ) -connected ( 2 n + 1 ) -dimensional differentiable manifold ( n ≥ 3 ) , then M admits a spinnable structure with axis S 2 n + 1 . Making use of the codimension-one foliation on S 2 n + 1 , this yields that M admits a codimension-foliation.

Foliations by planes and Lie group actions

J. A. Álvarez López, J. L. Arraut, C. Biasi (2003)

Annales Polonici Mathematici

Let N be a closed orientable n-manifold, n ≥ 3, and K a compact non-empty subset. We prove that the existence of a transversally orientable codimension one foliation on N∖K with leaves homeomorphic to ℝ n - 1 , in the relative topology, implies that K must be connected. If in addition one imposes some restrictions on the homology of K, then N must be a homotopy sphere. Next we consider C² actions of a Lie group diffeomorphic to ℝ n - 1 on N and obtain our main result: if K, the set of singular points of the...

Foliations of M 3 defined by ℝ 2 -actions

Jose Luis Arraut, Marcos Craizer (1995)

Annales de l'institut Fourier

In this paper we give a geometric characterization of the 2-dimensional foliations on compact orientable 3-manifolds defined by a locally free smooth action of ℝ 2 .

Foliations of surfaces I : an ideal boundary

John N. Mather (1982)

Annales de l'institut Fourier

Let F be a foliation of the punctured plane P . Any non-compact leaf of F has two ends, which we call leaf-ends. The set ℰ of leaf-ends which converge to the origin has a natural cyclic order. In the case ℰ is infinite, we show that the cyclicly ordered set β , obtained by identifying neighbors in ℰ and filling in the holes according to the Dedeking process, is equivalent to a circle. We show that the set P ∐ β has a natural topology, and it is homeomorphic to S 1 × [ 0 , ∞ ) with respect to this topology.

Foliations on the complex projective plane with many parabolic leaves

Marco Brunella (1994)

Annales de l'institut Fourier

We prove that a foliation on C P 2 with hyperbolic singularities and with “many" parabolic leaves (i.e. leaves without Green functions) is in fact a linear foliation. This is done in two steps: first we prove that there exists an algebraic leaf, using the technique of harmonic measures, then we show that the holonomy of this leaf is linearizable, from which the result follows easily.

Foliations with all leaves compact

D. B. A. Epstein (1976)

Annales de l'institut Fourier

The notion of the “volume" of a leaf in a foliated space is defined. If L is a compact leaf, then any leaf entering a small neighbourhood of L either has a very large volume, or a volume which is approximatively an integral multiple of the volume of L . If all leaves are compact there are three related objects to study. Firstly the topology of the quotient space obtained by identifying each leaf to a point ; secondly the holonomy of a leaf ; and thirdly whether the leaves have a locally bounded volume....

Foliations with complex leaves

Giuliana Gigante, Giuseppe Tomassini (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let X be a smooth foliation with complex leaves and let D be the sheaf of germs of smooth functions, holomorphic along the leaves. We study the ringed space X , D . In particular we concentrate on the following two themes: function theory for the algebra D ⁢ X and cohomology with values in D .

Formality and the Lefschetz property in symplectic and cosymplectic geometry

Giovanni Bazzoni, Marisa Fernández, Vicente Muñoz (2015)

Complex Manifolds

We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b1 = 1 case (in the coKähler/cosymplectic situation).

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