Holomorphic foliations in ruled surfaces

Gómez-Mont, Xavier

doi:10.1090/s0002-9947-1989-0983870-8

Cited by 9 publications

(4 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proof of Theorem 2 involves the local result of Theorem 1 and the unique continuation property established in Proposition 1, together with some global arguments relying on the Enriques-Kodaira classification of compact complex surfaces and structure results for non-singular holomorphic foliations [15,24] and authomorphism groups [39] of ruled surfaces.…”

Section: Theorem 2 a Compact Almost Kähler 4-manifold Whose Curvaturmentioning

confidence: 99%

“…By the previous argument, we may also assume that (M, I ) does not admit any elliptic fibration. According to the the classification of non-singular holomorphic foliations (see [15,Prop.6] and [24,Sec.3]), under the above assumptions for (M, I ), the following two cases arise: Case 1: F is tangent to the rational fibers; since (M, I ) is not an elliptic fibration, according to [39] we have H 0 (M, T F) = 0 as claimed.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…a Riccati foliation in the terminology of [24]. In this case c 1 (T F) is the pull-back of a class of H 2 (B, Z) (see [24]), so that c 1 (T F) is a multiple of the Poincaré-dual of any rational fiber F . Since [F ] · [F ] = 0 in homology and in view of (75), we have…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 2 more Smart Citations

Local models and integrability of certain almost Kähler 4-manifolds

Apostolov¹,

Armstrong²,

Drăghici³

2002

Mathematische Annalen

View full text Add to dashboard Cite

We classify, up to a local isometry, all non-Kähler almost Kähler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kähler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure. (2000): 53B20, 53C25 Mathematics Subject Classification

show abstract

Section: Theorem 2 a Compact Almost Kähler 4-manifold Whose Curvaturmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Local models and integrability of certain almost Kähler 4-manifolds

Apostolov¹,

Armstrong²,

Drăghici³

2002

Mathematische Annalen

View full text Add to dashboard Cite

show abstract

“…It would be interesting to investigate the existence of foliated projective structures on general foliated ruled surfaces (the work of Gómez-Mont [GM89] seems a natural starting point). Most foliations on these seem to have foliated affine structures.…”

Section: Definitions and The Problem Of Existencementioning

confidence: 99%

Foliated affine and projective structures

Deroin¹,

Guillot²

2023

Compositio Math.

View full text Add to dashboard Cite

We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces.

show abstract

Feuilletages holomorphes sur les surfaces complexes compactes

Brunella

1997

Annales Scientifiques de l’École Normale Supérieure

View full text Add to dashboard Cite

Feuilletages holomorphes sur les surfaces complexes compactes Annales scientifiques de l'É.N.S. 4 e série, tome 30, n o 5 (1997), p. 569-594 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1997, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

show abstract

Holomorphic foliations in ruled surfaces

Abstract: We analyse the universal families of holomorphic foliations with singularities in a ruled surface. In terms of Chern classes we determine the general and the special families. We also classify all nonsingular foliations.

Cited by 9 publications

References 10 publications

Local models and integrability of certain almost Kähler 4-manifolds

Local models and integrability of certain almost Kähler 4-manifolds

Foliated affine and projective structures

Feuilletages holomorphes sur les surfaces complexes compactes

Contact Info

Product

Resources

About