Partial holomorphic connections and extension of foliations
Isaia Nisoli
Source: Kyoto J. Math. Volume 52, Number 3
(2012), 517-555.
Abstract
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.
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Permanent link to this document: http://projecteuclid.org/euclid.kjm/1343309706
Digital Object Identifier: doi:10.1215/21562261-1625190
Zentralblatt MATH identifier: 06081383
Mathematical Reviews number (MathSciNet): MR2959947
References
[ABT1] M. Abate, F. Bracci, and F. Tovena, Index theorems for holomorphic self-maps, Ann. of Math. (2) 159 (2004), 819–864.
Mathematical Reviews (MathSciNet): MR2081441
Zentralblatt MATH: 1056.37025
Digital Object Identifier: doi:10.4007/annals.2004.159.819
[ABT2] M. Abate, F. Bracci, and F. Tovena, Index theorems for holomorphic maps and foliations, Indiana Univ. Math J. 57 (2008), 2999–3048.
Mathematical Reviews (MathSciNet): MR2492224
Zentralblatt MATH: 1179.32010
Digital Object Identifier: doi:10.1512/iumj.2008.57.3729
[ABT3] M. Abate, F. Bracci, and F. Tovena, Embeddings of submanifolds and normal bundles, Adv. Math. 220 (2009), 620–656.
Mathematical Reviews (MathSciNet): MR2466428
Zentralblatt MATH: 1161.32011
Digital Object Identifier: doi:10.1016/j.aim.2008.10.001
[Ati] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 182–207.
Mathematical Reviews (MathSciNet): MR86359
Zentralblatt MATH: 0078.16002
Digital Object Identifier: doi:10.1090/S0002-9947-1957-0086359-5
[Br] F. Bracci, First order extensions of holomorphic foliations, Hokkaido Math. J. 33 (2004), 473–490.
Mathematical Reviews (MathSciNet): MR2073011
Zentralblatt MATH: 1082.32021
Project Euclid: euclid.hokmj/1285766178
[Bru] M. Brunella, Some remarks on indices of holomorphic vector fields, Publ. Mat. 41 (1997), 527–544.
Mathematical Reviews (MathSciNet): MR1485502
Zentralblatt MATH: 0912.32024
Digital Object Identifier: doi:10.5565/PUBLMAT_41297_17
[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.
Mathematical Reviews (MathSciNet): MR1810535
Zentralblatt MATH: 0991.32020
Digital Object Identifier: doi:10.1090/conm/269/04328
[CL] C. Camacho and D. Lehmann, Residues of holomorphic foliations relative to a general submanifold, Bull. London Math. Soc. 37 (2005), 435–445.
Mathematical Reviews (MathSciNet): MR2131398
Zentralblatt MATH: 1088.32018
Digital Object Identifier: doi:10.1112/S0024609305004339
[CMS] C. Camacho, H. Movasati, and P. Sad, Fibered neighborhoods of curves in surfaces, J. Geom. Anal. 13 (2003), 55–66.
Mathematical Reviews (MathSciNet): MR1967036
Zentralblatt MATH: 1036.32009
Digital Object Identifier: doi:10.1007/BF02930996
[CS] C. Camacho and P. Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2) 115 (1982), 579–595.
Mathematical Reviews (MathSciNet): MR657239
Zentralblatt MATH: 0503.32007
Digital Object Identifier: doi:10.2307/2007013
[Ei] D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Grad. Texts in Math. 150, Springer, New York, 1995.
Mathematical Reviews (MathSciNet): MR1322960
[Gro] A. Grothendieck, A general theory of Fibre Spaces With Structure Sheaf, preprint, 2nd. ed., 1958, www.math.jussieu.fr/~leila/grothendieckcircle/GrothKansas.pdf
[Ho] T. Honda, Tangential index of foliations with curves on surfaces, Hokkaido Math. J. 33 (2004), 255–273.
Mathematical Reviews (MathSciNet): MR2072998
Zentralblatt MATH: 1067.37061
Project Euclid: euclid.hokmj/1285766165
[KS] B. Khanedani and T. Suwa, First variations of holomorphic forms and some applications, Hokkaido Math. J. 26 (1997), 323–335.
Mathematical Reviews (MathSciNet): MR1463088
Zentralblatt MATH: 0897.32013
[Lee] J. M. Lee, Introduction to Smooth Manifolds, Grad. Texts in Math. 218, Springer, New York, 2003.
Mathematical Reviews (MathSciNet): MR1930091
[LS1] D. Lehmann and T. Suwa, Residues of holomorphic vector fields relative to singular invariant subvarieties, J. Differential Geom. 42 (1995), 165–192.
Mathematical Reviews (MathSciNet): MR1350698
Zentralblatt MATH: 0844.32007
Project Euclid: euclid.jdg/1214457035
[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.
Mathematical Reviews (MathSciNet): MR1688141
Zentralblatt MATH: 1039.32041
Digital Object Identifier: doi:10.1142/S0129167X99000136
[MS] J. W. Milnor and J. Stasheff, Characteristic Classes, Princeton Univ. Press, Princeton, 1957.
Mathematical Reviews (MathSciNet): MR440554
[MY] Y. Mitera and J. Yoshizaki, The local analytical triviality of a complex analytic singular foliation, Hokkaido Math. J. 33 (2004), 275–297.
Mathematical Reviews (MathSciNet): MR2072999
Zentralblatt MATH: 1074.32011
Project Euclid: euclid.hokmj/1285766166
[Sa] K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 265–291.
Mathematical Reviews (MathSciNet): MR586450
[Su] T. Suwa, Indices of Vector Fields and Residues of Singular Holomorphic Foliations, Actualités Math., Hermann, Paris, 1998.
Mathematical Reviews (MathSciNet): MR1649358
Zentralblatt MATH: 0910.32035
Kyoto Journal of Mathematics
