scholarly journals Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves

2004 ◽  
Vol 356 (7) ◽  
pp. 2963-2988
Author(s):  
A. Lins Neto ◽  
J. C. Canille Martins
Keyword(s):  
Holomorphic Foliation ◽  
Hermitian Metrics ◽  
Poincaré Metric

PLURISUBHARMONIC VARIATION OF THE LEAFWISE POINCARÉ METRIC

2003 ◽  
Vol 14 (02) ◽  
pp. 139-151 ◽  
Author(s):  
MARCO BRUNELLA
Keyword(s):  
Kähler Manifold ◽  
Holomorphic Foliation ◽  
Poincaré Metric ◽  
One Dimensional ◽  
Kahler Manifold ◽  
Higher Dimensional ◽  
Compact Kähler Manifold

This paper is concerned with a higher dimensional generalization of the main result of our previous paper [2]: we shall prove that the Poincaré metric on the leaves of a one-dimensional holomorphic foliation on a compact Kähler manifold has a plurisubharmonic variation.

scholarly journals Quaternionic contact 4n + 3-manifolds and their 4n-quotients

2021 ◽  
Author(s):  
Yoshinobu Kamishima
Keyword(s):  
Scalar Curvature ◽  
Lie Group ◽  
Contact Structure ◽  
Einstein Manifolds ◽  
Hermitian Metrics ◽  
Kähler Metric ◽  
Formula One ◽  
Quaternion Space ◽  
Kahler Metric ◽  
Do So

AbstractWe study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics $$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$ ( g a , { J α } α = 1 3 ) on the domain Y of the standard quaternion space $${\mathbb {H}}^n$$ H n one of which, say $$(g_a,J_1)$$ ( g a , J 1 ) is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie group$${{\mathcal {M}}}$$ M to obtain quaternionic Hermitian metrics on the quotient Y of X by $${\mathbb {R}}^3$$ R 3 .

scholarly journals Generic pseudogroups on and the topology of leaves

2013 ◽  
Vol 149 (8) ◽  
pp. 1401-1430 ◽  
Author(s):  
J.-F. Mattei ◽  
J. C. Rebelo ◽  
H. Reis
Keyword(s):  
Invariant Curve ◽  
Holomorphic Foliation ◽  
Simply Connected

AbstractWe show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.

scholarly journals On the geometry of Poincaré's problem for one-dimensional projective foliations

2001 ◽  
Vol 73 (4) ◽  
pp. 475-482 ◽  
Author(s):  
MARCIO G. SOARES
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Projective Variety ◽  
Holomorphic Foliation ◽  
One Dimensional ◽  
Complex Projective Variety ◽  
Geometric Objects ◽  
Complex Projective

We consider the question of relating extrinsic geometric characters of a smooth irreducible complex projective variety, which is invariant by a one-dimensional holomorphic foliation on a complex projective space, to geometric objects associated to the foliation.

scholarly journals Invariance of basic Hodge numbers under deformations of Sasakian manifolds

2020 ◽  
Author(s):  
Paweł Raźny
Keyword(s):  
Elliptic Operators ◽  
Holomorphic Foliation ◽  
Sasakian Manifolds ◽  
Contact Manifolds ◽  
Continuity Theorem ◽  
Riemannian Foliations ◽  
Holomorphic Structure ◽  
Sasakian Structure ◽  
Hodge Numbers ◽  
Small Deformations

Abstract We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi-continuity theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with homologically orientable transversely Riemannian foliations. We use this to prove that the $$\partial {\bar{\partial }}$$ ∂ ∂ ¯ -lemma and being transversely Kähler are rigid properties under small deformations of the transversely holomorphic structure which preserve the foliation. We study an example which shows that this is not the case for arbitrary deformations of the transversely holomorphic foliation. Finally we point out an application of the upper-semi continuity theorem to K-contact manifolds.

On the Kähler-likeness on nearly Kähler manifolds

2020 ◽  
pp. 2150132
Author(s):  
Masaya Kawamura
Keyword(s):  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Hermitian Metrics ◽  
Kahler Manifold ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds ◽  
Nearly Kähler Manifold

We introduce Kähler-like, G-Kähler-like almost Hermitian metrics. We characterize the Kähler-likeness and the G-Kähler-likeness, and show that these properties are equivalent on nearly Kähler manifolds. Furthermore, we prove that a nearly Kähler manifold with the Kähler-likeness is Kähler.

Asymptotic Behavior of Normal Mappings of Several Complex Variables

1984 ◽  
Vol 36 (4) ◽  
pp. 718-746 ◽  
Author(s):  
Kyong T. Hahn
Keyword(s):  
Closed Subspace ◽  
Integral Formula ◽  
Holomorphic Mappings ◽  
Compact Open Topology ◽  
Hermitian Metrics ◽  
Easy Consequence ◽  
Normal Functions ◽  
Regular Sequences ◽  
Hermitian Manifolds ◽  
Continuous Mappings

Let M and N be connected Hermitian manifolds of dimensions m and n with Hermitian metrics hM and hN, respectively. Then the space ℓ(M, N) of continuous mappings between M and N endowed with the compact-open topology is second countable so that a metric can be furnished in ℓ(M, N) which induces the compact-open topology. A sequence {fn} in ℓ(M, N) converges to a n f in ℓ(M, N) in this topology if and only if fn converges to f uniformly on compact subsets of M. It is then an easy consequence of the Cauchy integral formula to show that the space ℋ(M, N) of holomorphic mappings f:M → N is a closed subspace of ℓ(M, N).In this paper, generalizing the classical notions of normal functions, Bloch functions, regular sequences and P-point sequences of one complex variable to the mappings in ℋ(M, N), see also [25], we obtain various relations which exist between these notions.

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