On Levi-flat hypersurfaces tangent to holomorphic webs

Arturo Fernández-Pérez[1]

  • [1] Departamento de Matemática, UFMG, Av. Antônio Carlos, 6627 C.P. 702, 30123-970 – Belo Horizonte – MG, Brazil.

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: 3, page 581-597
  • ISSN: 0240-2963

Abstract

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We investigate real analytic Levi-flat hypersurfaces tangent to holomorphic webs. We introduce the notion of first integrals for local webs. In particular, we prove that a k -web with finitely many invariant subvarieties through the origin tangent to a Levi-flat hypersurface has a holomorphic first integral.

How to cite

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Fernández-Pérez, Arturo. "On Levi-flat hypersurfaces tangent to holomorphic webs." Annales de la faculté des sciences de Toulouse Mathématiques 20.3 (2011): 581-597. <http://eudml.org/doc/219843>.

@article{Fernández2011,
abstract = {We investigate real analytic Levi-flat hypersurfaces tangent to holomorphic webs. We introduce the notion of first integrals for local webs. In particular, we prove that a $k$-web with finitely many invariant subvarieties through the origin tangent to a Levi-flat hypersurface has a holomorphic first integral.},
affiliation = {Departamento de Matemática, UFMG, Av. Antônio Carlos, 6627 C.P. 702, 30123-970 – Belo Horizonte – MG, Brazil.},
author = {Fernández-Pérez, Arturo},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Levi-flat hypersurfaces; first integrals; local webs},
language = {eng},
month = {7},
number = {3},
pages = {581-597},
publisher = {Université Paul Sabatier, Toulouse},
title = {On Levi-flat hypersurfaces tangent to holomorphic webs},
url = {http://eudml.org/doc/219843},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Fernández-Pérez, Arturo
TI - On Levi-flat hypersurfaces tangent to holomorphic webs
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/7//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - 3
SP - 581
EP - 597
AB - We investigate real analytic Levi-flat hypersurfaces tangent to holomorphic webs. We introduce the notion of first integrals for local webs. In particular, we prove that a $k$-web with finitely many invariant subvarieties through the origin tangent to a Levi-flat hypersurface has a holomorphic first integral.
LA - eng
KW - Levi-flat hypersurfaces; first integrals; local webs
UR - http://eudml.org/doc/219843
ER -

References

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  2. Brunella (M.).— Singular Levi-flat hypersurfaces and codimension one foliations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6, no. 4, p. 661-672 (2007). Zbl1214.32012MR2394414
  3. Cavalier (V.), Lehmann (D.).— Introduction à l’étude globale des tissus sur une surface holomorphe. Ann. Inst. Fourier (Grenoble) 57, no. 4, p. 1095-1133 (2007). Zbl1129.14015MR2339328
  4. Cavalier (V), Lehmann (D.).— Global structure of holomorphic webs on surfaces. Geometry and topology of caustics-CAUSTICS ’06, 35-44, Banach Center Publ., 82, Polish Acad. Sci. Inst. Math., Warsaw (2008). Zbl1149.14300MR2463596
  5. Cerveau (D.), Lins Neto (A.).— Local Levi-flat hypersurfaces invariants by a codimension one holomorphic foliation, To appear in Amer. J. Math. Zbl1225.32038
  6. Fernández-Pérez (A.).— On normal forms of singular Levi-flat real analytic hypersurfaces. Bull. Braz. Math. Soc. (N.S.) 42, no. 1, p. 75-85 (2011). Zbl1215.32016MR2774175
  7. Fernández-Pérez (A.).— Singular Levi-flat hypersurfaces. An approach through holomorphic foliations. Ph.D Thesis IMPA, Brazil (2010). 
  8. Gunning (R.).— Introduction to holomorphic functions of several variables. Vol. II. Local theory. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA (1990). MR1057177
  9. Lebl (J.).— Singularities and complexity in CR geometry. Ph.D. Thesis, University of California at San Diego, Spring (2007). MR2709940
  10. Loray (F.).— Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. ; http://hal.archives-ouvertures.fr/ccsd-00016434 
  11. Mattei (J.F.), Moussu (R.).— Holonomie et intégrales premières, Ann. Ec. Norm. Sup. 13, p. 469-523 ( 1980 ) . Zbl0458.32005MR608290
  12. Pereira (J.V.), Pirio (L.).— An invitation to web geometry. From Abel’s addition theorem to the algebraization of codimension one webs. Publicações Matemáticas do IMPA. Rio de Janeiro (2009). Zbl1184.53002MR2536234
  13. Seidenberg (A.).— Reduction of singularities of the differential equation A d y = B d x . Amer. J. Math. 90, p. 248-269 ( 1968 ) . Zbl0159.33303MR220710
  14. Yartey (J.N.A).— Number of singularities of a generic web on the complex projective plane. J. Dyn. Control Syst. 11, no. 2, p. 281-296 (2005). Zbl1066.37032MR2131813
  15. Siu (Y.T.).— Techniques of extension of analytic objects. Lecture Notes in Pure and Applied Mathematics, Vol. 8. Marcel Dekker, Inc., New York (1974). Zbl0294.32007MR361154

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