Advertisement

Infinitesimal deformations of Levi flat hypersurfaces

  • Andrei IordanEmail author
Article
  • 24 Downloads

Abstract

In order to study the deformations of foliations of codimension 1 of a smooth manifold L, de Bartolomeis and Iordan defined the DGLA \( \mathcal {Z}^{*}\left( L\right) \), where \(\mathcal {Z}^{*}\left( L\right) \) is a subset of differential forms on L. In another paper, de Bartolomeis and Iordan studied the deformations of foliations of a smooth manifold L by defining the canonical solutions of Maurer–Cartan equation in the DGLA of graded derivations \(\mathcal {D}^{*}\left( L\right) \). Let L be a Levi flat hypersurface in a complex manifold. Then the deformation theories in \(\mathcal {Z}^{*}\left( L\right) \) and \(\mathcal {D }^{*}\left( L\right) \) lead to the moduli space for the Levi flat deformations of L. In this paper we discuss the relationship between the infinitesimal deformations of L defined by the solutions of Maurer–Cartan equation in \(\mathcal {Z}^{*}\left( L\right) \) and the infinitesimal deformations of L obtained by means of the canonical solutions of Maurer–Cartan equation in the DGLA of graded derivations \(\mathcal {D}^{*}\left( L\right) \).

Keywords

Levi flat hypersurface Differential graded Lie Algebras Maurer–Cartan equation Foliations Graded derivations 

Mathematics Subject Classification

Primary 32G10 32E99 16W25 53C12 

References

  1. 1.
    de Bartolomeis, P., Iordan, A.: Deformations of Levi flat hypersurfaces in complex manifolds. Ann. Scient. Éc. Norm. Sup. 48(2), 281–311 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    de Bartolomeis, P., Iordan, A.: Maurer–Cartan equation in the DGLA of graded derivations, (2015), Preprint arXiv:1506.06732
  3. 3.
    de Bartolomeis, P., Iordan, A.: On the obstruction of the deformation theory in the DGLA of graded derivations, Complex and Symplectic Geometry (D. Angella, C. Medori and A. Tomassini, ed.), vol. 21, Springer Indam Series, pp. 95–105 (2017)Google Scholar
  4. 4.
    Frölicher, A., Nijenhuis, A.: Theory of vector valued differential forms. Part I. Derivations of the graded ring of differential forms. Indag. Math. 18, 338–359 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Godbillon, C.: Feuilletages: Etudes géométriques. Progress in Mathematics, Birkhäuser, Basel (1991)zbMATHGoogle Scholar
  6. 6.
    Kodaira, K.: On deformations of some complex pseudo-group structures. Ann. Math. 71, 224–302 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kodaira, K., Spencer, D.: Multifoliate structures. Ann. Math. 74(1), 52–100 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Michor, P.W.: Topics in differential geometry. AMS, Providence (2008)zbMATHGoogle Scholar
  9. 9.
    Spencer, D.C.: Some remarks on perturbation of structure, Analytic Functions, pp. 67–87. Princeton University Press, Princeton, NJ (1960)Google Scholar

Copyright information

© Unione Matematica Italiana 2018

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu-Paris Rive GaucheFaculté des Sciences et Ingénierie (former Université Pierre et Marie Curie), Sorbonne UniversitéParis Cedex 05France

Personalised recommendations