scholarly journals Equivariant embeddings of strongly pseudoconvex Cauchy–Riemann manifolds

2021 ◽  
Author(s):  
Kevin Fritsch ◽  
Peter Heinzner
Keyword(s):  
Complex Manifold ◽  
Lie Group ◽  
Complex Space ◽  
Complex Geometry ◽  
Embedding Theorem ◽  
Cr Manifold ◽  
Projective Embedding ◽  
Cr Manifolds ◽  
Riemann Manifolds ◽  
Cr Map

AbstractLet X be a CR manifold with transversal, proper CR action of a Lie group G. We show that the quotient X/G is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorizes uniquely over a holomorphic map on X/G. We then use this result and complex geometry to prove an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal $$G \rtimes S^1$$ G ⋊ S 1 -action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.

scholarly journals A note on para-holomorphic Riemannian–Einstein manifolds

2016 ◽  
Vol 13 (09) ◽  
pp. 1650107
Author(s):  
Cristian Ida ◽  
Alexandru Ionescu ◽  
Adelina Manea
Keyword(s):  
Complex Manifold ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Complex Geometry ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Einstein Metric ◽  
Einstein Manifolds ◽  
Norden Metric ◽  
Norden Metrics

The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. First, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using a one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler–Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kähler–Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits a natural para-Kählerian–Norden Einstein metric.

scholarly journals SYMPLECTIC STABILITY, ANALYTIC STABILITY IN NON-ALGEBRAIC COMPLEX GEOMETRY

2004 ◽  
Vol 15 (02) ◽  
pp. 183-209 ◽  
Author(s):  
ANDREI TELEMAN
Keyword(s):  
Weight Function ◽  
Complex Manifold ◽  
Stability Theory ◽  
Complex Geometry ◽  
Reductive Group ◽  
Hamiltonian Action ◽  
Maximal Weight ◽  
Analytic Stability ◽  
The Stability

We give a systematic presentation of the stability theory in the non-algebraic Kählerian geometry. We introduce the concept of "energy complete Hamiltonian action". To an energy complete Hamiltonian action of a reductive group G on a complex manifold one can associate a G-equivariant maximal weight function and prove a Hilbert criterion for semistability. In other words, for such actions, the symplectic semistability and analytic semistability conditions are equivalent.

scholarly journals On The Holomorphic Automorphism Groups of Complex Spaces

1968 ◽  
Vol 33 ◽  
pp. 85-106 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Lie Group ◽  
Complex Space ◽  
Automorphism Groups ◽  
Analogous Result ◽  
Holomorphic Mapping ◽  
Compact Complex Manifold ◽  
Holomorphic Automorphism ◽  
Complex Spaces ◽  
Weaker Conditions ◽  
Holomorphic Automorphism Groups

For a complex space X we consider the group Aut (X) of all automorphisms of X, where an automorphism means a holomorphic automorphism, i.e. an injective holomorphic mapping of X onto X itself with the holomorphic inverse. In 1935, H. Cartan showed that Aut (X) has a structure of a real Lie group if X is a bounded domain in CN([7]) and, in 1946, S. Bochner and D. Montgomery got the analogous result for a compact complex manifold X ([2] and [3]). Afterwards, the latter was generalized by R.C. Gunning ([11]) and H. Kerner ([16]), and the former by W. Kaup ([14]), to complex spaces. The purpose of this paper is to generalize these results to the case of complex spaces with weaker conditions. For brevity, we restrict ourselves to the study of σ-compact irreducible complex spaces only.

scholarly journals Regularity of CR mappings of abstract CR structures

2019 ◽  
Vol 31 (01) ◽  
pp. 2050009
Author(s):  
Bernhard Lamel ◽  
Nordine Mir
Keyword(s):  
Euclidean Space ◽  
Infinite Order ◽  
Extension Property ◽  
Cr Manifold ◽  
Generic Point ◽  
Cr Mappings ◽  
Complex Euclidean Space ◽  
Cr Structures ◽  
Cr Map ◽  
Regularity Results

We study the [Formula: see text] regularity problem for CR maps from an abstract CR manifold [Formula: see text] into some complex Euclidean space [Formula: see text]. We show that if [Formula: see text] satisfies a certain condition called the microlocal extension property, then any [Formula: see text]-smooth CR map [Formula: see text], for some integer [Formula: see text], which is nowhere [Formula: see text]-smooth on some open subset [Formula: see text] of [Formula: see text], has the following property: for a generic point [Formula: see text] of [Formula: see text], there must exist a formal complex subvariety through [Formula: see text], tangent to [Formula: see text] to infinite order, and depending in a [Formula: see text] and CR manner on [Formula: see text]. As a consequence, we obtain several [Formula: see text] regularity results generalizing earlier ones by Berhanu–Xiao and the authors (in the embedded case).

Complex geometry of space–time motivated by gravity’s rainbow

2019 ◽  
Vol 97 (5) ◽  
pp. 558-561
Author(s):  
Faizan Bhat ◽  
Mussadiq H. Qureshi ◽  
Manzoor A. Malik ◽  
Asif Iqbal
Keyword(s):  
Imaginary Part ◽  
Complex Space ◽  
Complex Geometry ◽  
Planck Scale ◽  
Space Time ◽  
Low Energy ◽  
Gravity’S Rainbow ◽  
Gravity's Rainbow ◽  
Energy Approximation ◽  
Planck Energy

In this paper, we generalize the formalism of gravity’s rainbow to complex space–time. The resulting geometry depends on the energy of the probe in such a way that the usual real manifold is the low energy approximation of the Planck scale geometry of space–time. So, our formalism agrees with all the observational data about our space–time being real, as at the scale these experiments are preformed, the imaginary part of the geometry is suppressed by Planck energy. However, the imaginary part of the geometry becomes important near the Planck energy, and so it cannot be neglected near the Planck scale. So, the Planck scale geometry of space–time is described by a complex manifold.

scholarly journals The asymptotics of the analytic torsion on CR manifolds with S1 action

2019 ◽  
Vol 21 (04) ◽  
pp. 1750094 ◽  
Author(s):  
Chin-Yu Hsiao ◽  
Rung-Tzung Huang
Keyword(s):  
Line Bundle ◽  
Asymptotic Formula ◽  
Analytic Torsion ◽  
Cr Manifold ◽  
Cr Manifolds ◽  
Dimension Formula ◽  
Positive Line Bundle ◽  
Positive Line

Let [Formula: see text] be a compact connected strongly pseudoconvex CR manifold of dimension [Formula: see text], [Formula: see text] with a transversal CR [Formula: see text]-action on [Formula: see text]. We introduce the Fourier components of the Ray–Singer analytic torsion on [Formula: see text] with respect to the [Formula: see text]-action. We establish an asymptotic formula for the Fourier components of the analytic torsion with respect to the [Formula: see text]-action. This generalizes the asymptotic formula of Bismut and Vasserot on the holomorphic Ray–Singer torsion associated with high powers of a positive line bundle to strongly pseudoconvex CR manifolds with a transversal CR [Formula: see text]-action.

CR Mappings of Circular CR Manifolds

1995 ◽  
Vol 38 (4) ◽  
pp. 396-407
Author(s):  
André Boivin ◽  
Roman Dwilewicz
Keyword(s):  
Finite Number ◽  
Type Equation ◽  
Vector Spaces ◽  
Cr Manifold ◽  
Complex Vector ◽  
Cr Manifolds ◽  
Cr Mappings

AbstractLet M be a circular CR manifold and let N be a rigid CR manifold in some complex vector spaces. The problem of the existence of local CR mappings from M into N is considered. Conditions are given which ensure that the space of such CR mappings depends on a finite number of parameters. The idea of the proof of the main result relies on a Bishop type equation for CR mappings. Roughly speaking, we look for CR mappings from M into N in the form F = (ƒ,g), we assume that g is given, then we find ƒ in terms of g and some parameters, and finally we look for conditions on g. It works independently of assumptions on the Levi forms of M and N, and there is also some freedom on the codimension of the manifolds.

scholarly journals Analytic Jet Parametrization for CR Automorphisms of Some Essentially Finite CR Manifolds

2008 ◽  
Vol 189 ◽  
pp. 155-168
Author(s):  
Sung-Yeon Kim
Keyword(s):  
Uniform Convergence ◽  
Lie Group ◽  
Group Structure ◽  
Cr Manifolds ◽  
Real Analytic ◽  
The Stability

AbstractIn this paper we construct analytic jet parametrizations for the germs of real analytic CR automorphisms of some essentially finite CR manifolds on their finite jet at a point. As an application we show that the stability groups of such CR manifolds have Lie group structure under composition with the topology induced by uniform convergence on compacta.

scholarly journals On the Automorphism Group of a Holomorphic Fiber Bundle over A Complex Space

1970 ◽  
Vol 37 ◽  
pp. 91-106 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Automorphism Group ◽  
Lie Group ◽  
Fiber Bundle ◽  
Complex Space ◽  
Automorphism Groups ◽  
Compact Complex Manifold ◽  
Compact Open Topology ◽  
Complex Spaces ◽  
Holomorphic Fiber Bundle ◽  
Holomorphic Automorphisms

In [8], A. Morimoto proved that the automorphism group of a holomorphic principal fiber bundle over a compact complex manifold has a structure of a complex Lie group with the compact-open topology. The purpose of this paper is to get similar results on the automorphism groups of more general types of locally trivial fiber spaces over complex spaces. We study automorphisms of a holomorphic fiber bundle over a complex space which has a complex space Y as the fiber and a (not necessarily complex Lie) group G of holomorphic automorphisms of Y as the structure group (see Definition 3. l).

scholarly journals THE PICARD GROUP OF THE LOOP SPACE OF THE RIEMANN SPHERE

2010 ◽  
Vol 21 (11) ◽  
pp. 1387-1399
Author(s):  
NING ZHANG
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Loop Space ◽  
Riemann Sphere ◽  
Loop Group ◽  
Picard Group ◽  
Line Bundles ◽  
Projective Embedding ◽  
Infinite Dimensional ◽  
Holomorphic Line Bundles

The loop space Lℙ1 of the Riemann sphere consisting of all Ck or Sobolev Wk, p maps S1 → ℙ1 is an infinite dimensional complex manifold. We compute the Picard group pic(Lℙ1) of holomorphic line bundles on Lℙ1 as an infinite dimensional complex Lie group with Lie algebra the Dolbeault group H0, 1(Lℙ1). The group G of Möbius transformations and its loop group LG act on Lℙ1. We prove that an element of pic(Lℙ1) is LG-fixed if it is G-fixed, thus completely answering the question of Millson and Zombro about the G-equivariant projective embedding of Lℙ1.

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