Kyoto Journal of Mathematics

Partial holomorphic connections and extension of foliations

Isaia Nisoli

Full-text: Open access


This paper stresses the strong link between the existence of partial holomorphic connections on the normal bundle of a foliation seen as a quotient of the ambient tangent bundle and the extendability of a foliation to an infinitesimal neighborhood of a submanifold. We find the obstructions to extendability, and thanks to the theory developed we obtain some new Khanedani–Lehmann–Suwa type index theorems.

Article information

Kyoto J. Math., Volume 52, Number 3 (2012), 517-555.

First available in Project Euclid: 26 July 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37F75: Holomorphic foliations and vector fields [See also 32M25, 32S65, 34Mxx]
Secondary: 32S65: Singularities of holomorphic vector fields and foliations 32A27: Local theory of residues [See also 32C30]


Nisoli, Isaia. Partial holomorphic connections and extension of foliations. Kyoto J. Math. 52 (2012), no. 3, 517--555. doi:10.1215/21562261-1625190.

Export citation


  • [ABT1] M. Abate, F. Bracci, and F. Tovena, Index theorems for holomorphic self-maps, Ann. of Math. (2) 159 (2004), 819–864.
  • [ABT2] M. Abate, F. Bracci, and F. Tovena, Index theorems for holomorphic maps and foliations, Indiana Univ. Math J. 57 (2008), 2999–3048.
  • [ABT3] M. Abate, F. Bracci, and F. Tovena, Embeddings of submanifolds and normal bundles, Adv. Math. 220 (2009), 620–656.
  • [Ati] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 182–207.
  • [Br] F. Bracci, First order extensions of holomorphic foliations, Hokkaido Math. J. 33 (2004), 473–490.
  • [Bru] M. Brunella, Some remarks on indices of holomorphic vector fields, Publ. Mat. 41 (1997), 527–544.
  • [Ca] C. Camacho, “Dicritical singularities of holomorphic vector fields” in Laminations and Foliations in Dynamics, Geometry and Topology (Stony Brook, N.Y., 1998), Contemp. Math. 269, Amer. Math. Soc., Providence, 2001, 39–45.
  • [CL] C. Camacho and D. Lehmann, Residues of holomorphic foliations relative to a general submanifold, Bull. London Math. Soc. 37 (2005), 435–445.
  • [CMS] C. Camacho, H. Movasati, and P. Sad, Fibered neighborhoods of curves in surfaces, J. Geom. Anal. 13 (2003), 55–66.
  • [CS] C. Camacho and P. Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2) 115 (1982), 579–595.
  • [Ei] D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Grad. Texts in Math. 150, Springer, New York, 1995.
  • [Gro] A. Grothendieck, A general theory of Fibre Spaces With Structure Sheaf, preprint, 2nd. ed., 1958,
  • [Ho] T. Honda, Tangential index of foliations with curves on surfaces, Hokkaido Math. J. 33 (2004), 255–273.
  • [KS] B. Khanedani and T. Suwa, First variations of holomorphic forms and some applications, Hokkaido Math. J. 26 (1997), 323–335.
  • [Lee] J. M. Lee, Introduction to Smooth Manifolds, Grad. Texts in Math. 218, Springer, New York, 2003.
  • [LS1] D. Lehmann and T. Suwa, Residues of holomorphic vector fields relative to singular invariant subvarieties, J. Differential Geom. 42 (1995), 165–192.
  • [LS2] D. Lehmann and T. Suwa, Generalization of variations and Baum–Bott residues for holomorphic foliations on singular varieties, Internat. J. Math. 10 (1999), 367–384.
  • [MS] J. W. Milnor and J. Stasheff, Characteristic Classes, Princeton Univ. Press, Princeton, 1957.
  • [MY] Y. Mitera and J. Yoshizaki, The local analytical triviality of a complex analytic singular foliation, Hokkaido Math. J. 33 (2004), 275–297.
  • [Sa] K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 265–291.
  • [Su] T. Suwa, Indices of Vector Fields and Residues of Singular Holomorphic Foliations, Actualités Math., Hermann, Paris, 1998.