A Survey on Levi Flat Hypersurfaces

Ohsawa, Takeo

doi:10.1007/978-3-319-20337-9_9

Cited by 7 publications

(5 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…So ρ is a strictly psh exhaustion function. [13,14], or in the families described by Ohsawa [24]. When M is infinitesimally homogeneous manifold [16,17], the domain U has a psh exhaustion function, but may not have strictly psh functions and hence is not necessarily Stein.…”

Section: A Condition For Steiness Of a Pseudoconvex Domainmentioning

confidence: 99%

“…When U admits a continuous psh exhaustion function, the domain U may not have strictly psh functions and hence is not necessarily Stein, this is the case in Grauert examples [13,14], or in the families described by Ohsawa [24]. When M is infinitesimally homogeneous manifold [16,17], the domain U has a psh exhaustion function, but may not have strictly psh functions and hence is not necessarily Stein.…”

Section: An Obstruction To Steiness: Levi Currentsmentioning

confidence: 99%

“…The Levi problem admits a positive solution in many cases, in particular, when M is C n or P n . We refer to the surveys by Narasimhan [22], Siu [29], Peternell [25] and to the recent discussion by Ohsawa [24]. See also the book by Hörmander [18].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Levi problem in complex manifolds

Sibony

2017

Math. Ann.

View full text Add to dashboard Cite

Let U be a pseudoconvex open set in a complex manifold M . When is U a Stein manifold? There are classical counter examples due to Grauert, even when U has real-analytic boundary or has strictly pseudoconvex points. We give new criteria for the Steinness of U and we analyze the obstructions. The main tool is the notion of Levi-currents. They are positive ∂∂-closed currents T of bidimension (1, 1) and of mass 1 directed by the directions where all continuous psh functions in U have vanishing Leviform. The extremal ones, are supported on the sets where all continuous psh functions are constant. We also construct under geometric conditions, bounded strictly psh exhaustion functions, and hence we obtain DonnellyFefferman weights. To any infinitesimally homogeneous manifold, we associate a foliation. The dynamics of the foliation determines the solution of the Levi-problem. Some of the results can be extended to the context of pseudoconvexity with respect to a Pfaff-system. Classification AMS 2010: Primary: 32Q28, 32U10; 32U40; 32W05; Secondary 37F75

show abstract

Section: A Condition For Steiness Of a Pseudoconvex Domainmentioning

confidence: 99%

Section: An Obstruction To Steiness: Levi Currentsmentioning

confidence: 99%

See 1 more Smart Citation

Levi problem in complex manifolds

Sibony

2017

Math. Ann.

View full text Add to dashboard Cite

show abstract

“…Lins Neto [18] a montré l'inexistence d'hypersurfaces Levi-plates analytiques réelles dans l'espace projectif complexe P n (C) (n 3). Citons aussi par exemple l'article de survol [20], et les articles [3], [12], [17], [9], [7] et [1] pour diverses constructions et résultats d'inexistence. Notons en particulier [19] qui construit des exemples différents de ceux que nous allons étudier dans les surfaces de Kummer.…”

Section: Introductionunclassified

Presque toute surface K3 contient une infinité d'hypersurfaces Levi-plates linéaires

Lequen¹

2021

Preprint

View full text Add to dashboard Cite

On s'intéresse à la construction d'hypersurfaces Levi-plates analytiques réelles dans les surfaces K3. On peut en construire dans les tores complexes de dimension 2 en prenant des images d'hyperplans réels. On montre que ≪ presque toute ≫ surface K3 contient une infinité d'hypersurfaces Levi-plates de ce type. La preuve repose principalement sur une construction récente due à Koike-Uehara, ainsi que sur les idées de Verbitsky sur les structures complexes ergodiques et une adaptation d'un argument dû à Ghys dans le cadre de l'étude de la topologie des feuilles génériques.

show abstract

“…Levi-flat hypersurfaces were studied in both fields of several complex analysis and holomorphic foliations. One of the origin is in the works for the Levi problem due to Grauert (See [Oh1], [Oh2] and references therein). On the other hand, the study has been developing in connection with the exceptional minimal set conjecture ( [CLS], [BLM] and [C]).…”

Section: Introductionmentioning

confidence: 99%