scholarly journals Rigidity properties of holomorphic Legendrian singularities

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Jun-Muk Hwang
Keyword(s):  
Line Bundle ◽  
Contact Manifold ◽  
Tangent Cones ◽  
Contact Manifolds ◽  
Holomorphic Family ◽  
Rigidity Results ◽  
Holomorphic Sections ◽  
Minor Revision

We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones. This result is partly motivated by a problem on Fano contact manifolds. The second result is the deformation-rigidity of normal Legendrian singularities, meaning that any holomorphic family of normal Legendrian singularities is trivial, up to contactomorphic biholomorphisms of germs. Both results are proved by exploiting the relation between infinitesimal contactomorphisms and holomorphic sections of the natural line bundle on the contact manifold. Comment: 21 pages, minor revision

scholarly journals Splitting theorem for sheaves of holomorphic k-vectors on complex contact manifolds

2018 ◽  
Vol 29 (13) ◽  
pp. 1850091
Author(s):  
Takayuki Moriyama ◽  
Takashi Nitta
Keyword(s):  
Line Bundle ◽  
Exact Sequence ◽  
Contact Structure ◽  
Integrability Condition ◽  
Contact Manifold ◽  
Splitting Theorem ◽  
Vanishing Theorems ◽  
Contact Manifolds ◽  
Structure Formula ◽  
Complex Contact Manifold

A complex contact structure [Formula: see text] is defined by a system of holomorphic local 1-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle [Formula: see text] of the tangent bundle and a line bundle [Formula: see text]. In this paper, we prove that the sheaf of holomorphic [Formula: see text]-vectors on a complex contact manifold splits into the sum of [Formula: see text] and [Formula: see text] as sheaves of [Formula: see text]-module. The theorem induces the short exact sequence of cohomology of holomorphic [Formula: see text]-vectors, and we obtain vanishing theorems for the cohomology of [Formula: see text].

Ricci almost solitons and contact geometry

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Amalendu Ghosh
Keyword(s):  
Vector Field ◽  
Contact Manifold ◽  
Ricci Soliton ◽  
Jacobi Field ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Rigidity Results ◽  
Parallel Ricci Tensor ◽  
Potential Vector Field

Abstract We prove that on a K-contact manifold, a Ricci almost soliton is a Ricci soliton if and only if the potential vector field V is a Jacobi field along the Reeb vector field ξ. Then we study contact metric as a Ricci almost soliton with parallel Ricci tensor. To this end, we consider Ricci almost solitons whose potential vector field is a contact vector field and prove some rigidity results.

scholarly journals Open books and exact symplectic cobordisms

2018 ◽  
Vol 29 (04) ◽  
pp. 1850026 ◽  
Author(s):  
Mirko Klukas
Keyword(s):  
Disjoint Union ◽  
Contact Manifold ◽  
Higher Dimensions ◽  
Fiber Bundles ◽  
Open Book ◽  
Contact Manifolds ◽  
Open Books ◽  
Symplectic Cobordism ◽  
The Given ◽  
Symplectic Cobordisms

Given two open books with equal pages, we show the existence of an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds associated to the given open books, and whose positive end induces the contact manifold associated to the open book with the same page and concatenated monodromy. Using similar methods, we show the existence of strong fillings for contact manifolds associated with doubled open books, a certain class of fiber bundles over the circle obtained by performing the binding sum of two open books with equal pages and inverse monodromies. From this we conclude, following an outline by Wendl, that the complement of the binding of an open book cannot contain any local filling obstruction. Given a contact [Formula: see text]-manifold, according to Eliashberg there is a symplectic cobordism to a fibration over the circle with symplectic fibers. We extend this result to higher dimensions recovering a recent result by Dörner–Geiges–Zehmisch. Our cobordisms can also be thought of as the result of the attachment of a generalized symplectic [Formula: see text]-handle.

scholarly journals A note on Berezin-Toeplitz quantization of the Laplace operator

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Alberto Della Vedova
Keyword(s):  
Line Bundle ◽  
Laplace Operator ◽  
Adjoint Operator ◽  
Holomorphic Sections ◽  
The Laplace Operator ◽  
Hodge Manifold

AbstractGiven a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.

On the Asymptotic Behavior of Complete Kähler Metrics of Positive Ricci Curvature

2010 ◽  
Vol 62 (1) ◽  
pp. 3-18
Author(s):  
Boudjemâa Anchouche
Keyword(s):  
Asymptotic Behavior ◽  
Line Bundle ◽  
Ricci Curvature ◽  
Projective Variety ◽  
Volume Form ◽  
Positive Ricci Curvature ◽  
Standard Type ◽  
Number Of Components ◽  
Holomorphic Sections ◽  
Volume Forms

AbstractLet (X, g) be a complete noncompact Kähler manifold, of dimension n ≥ 2, with positive Ricci curvature and of standard type (see the definition below). N. Mok proved that X can be compactified, i.e., X is biholomorphic to a quasi-projective variety. The aim of this paper is to prove that the L2 holomorphic sections of the line bundle K−qXand the volume form of the metric g have no essential singularities near the divisor at infinity. As a consequence we obtain a comparison between the volume forms of the Kähler metric g and of the Fubini-Study metric induced on X. In the case of dimC X = 2, we establish a relation between the number of components of the divisor D and the dimension of the.

scholarly journals Certain results on generalized (k,\mu)-contact manifolds

2017 ◽  
Vol 37 (3) ◽  
pp. 131
Author(s):  
Pradip Majhi ◽  
Gopal Ghosh
Keyword(s):  
Contact Manifold ◽  
Contact Manifolds

The object of the present paper is to study generalized (k,\mu)-contact manifolds. At first we consider \phi-semisymmetric generalized (k,\mu)-contact manifolds. Beside these we study extended pseudo projective at generalized (k,\mu)-contact manifolds. Also (k,\mu )-contact manifold satisfying \barP^e.S = 0 is also considered. As a consequence we obtain several corollaries.

scholarly journals Space time manifolds and contact structures

1990 ◽  
Vol 13 (3) ◽  
pp. 545-553 ◽  
Author(s):  
K. L. Duggal
Keyword(s):  
Contact Structure ◽  
Contact Manifold ◽  
Contact Structures ◽  
Timelike Vector ◽  
Contact Manifolds ◽  
New Class ◽  
Dimensional Spacetime ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes ◽  
Regular Contact

A new class of contact manifolds (carring a global non-vanishing timelike vector field) is introduced to establish a relation between spacetime manifolds and contact structures. We show that odd dimensional strongly causal (in particular, globally hyperbolic) spacetimes can carry a regular contact structure. As examples, we present a causal spacetime with a non regular contact structure and a physical model [Gödel Universe] of Homogeneous contact manifold. Finally, we construct a model of 4-dimensional spacetime of general relativity as a contact CR-submanifold.

scholarly journals Translated points on hypertight contact manifolds

2018 ◽  
Vol 10 (02) ◽  
pp. 289-322
Author(s):  
Matthias Meiwes ◽  
Kathrin Naef
Keyword(s):  
Floer Homology ◽  
Contact Manifold ◽  
Contact Form ◽  
Contact Manifolds ◽  
Rabinowitz Floer Homology

A contact manifold admitting a supporting contact form without contractible Reeb orbits is called hypertight. In this paper we construct a Rabinowitz–Floer homology associated to an arbitrary supporting contact form for a hypertight contact manifold [Formula: see text], and use this to prove versions of a conjecture of Sandon [17] on the existence of translated points and to show that positive loops of contactomorphisms give rise to non-contractible Reeb orbits.

scholarly journals Two closed orbits for non-degenerate Reeb flows

2020 ◽  
pp. 1-36
Author(s):  
MIGUEL ABREU ◽  
JEAN GUTT ◽  
JUNGSOO KANG ◽  
LEONARDO MACARINI
Keyword(s):  
Contact Manifold ◽  
Contact Manifolds ◽  
Symplectic Homology ◽  
Connected Sums ◽  
Reeb Flows ◽  
Closed Orbits ◽  
Finite Quotient ◽  
Closed Contact ◽  
First Chern Class ◽  
Toric Contact Manifolds

Abstract We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

scholarly journals Contact homology of good toric contact manifolds

2011 ◽  
Vol 148 (1) ◽  
pp. 304-334 ◽  
Author(s):  
Miguel Abreu ◽  
Leonardo Macarini
Keyword(s):  
Homotopy Class ◽  
Chern Class ◽  
Einstein Metrics ◽  
Contact Manifold ◽  
Infinite Family ◽  
Contact Structures ◽  
Contact Homology ◽  
Contact Manifolds ◽  
First Chern Class ◽  
Toric Contact Manifolds

AbstractIn this paper we show that any good toric contact manifold has a well-defined cylindrical contact homology, and describe how it can be combinatorially computed from the associated moment cone. As an application, we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of Gauntlett et al. on Sasaki–Einstein metrics. We show in particular that these give rise to a new infinite family of non-equivalent contact structures on S2×S3 in the unique homotopy class of almost contact structures with vanishing first Chern class.

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