Formal classification of two-dimensional neighborhoods of genus g2 curves with trivial normal bundle  [ Classification formelle des voisinages de courbes de genre g2 avec fibré normal trivial ]
Annales de l'Institut Fourier, à paraître, 22 p.

On étudie dans ce papier la classification formelle des voisinages de dimension deux de courbes de genre g2 dont le fibré normal est trivial. On construit tout d’abord sur de tels voisinages des feuilletages formels dont l’holonomie s’annule le long de nombreux lacets, puis on donne la classification formelle / analytique des voisinages équipés de deux feuilletages, et finalement on rassemble tout cela pour obtenir une description de l’espace des voisinages modulo équivalence formelle.

In this paper we study the formal classification of two-dimensional neighborhoods of genus g2 curves with trivial normal bundle. We first construct formal foliations on such neighborhoods with holonomy vanishing along many loops, then give the formal / analytic classification of neighborhoods equipped with two foliations, and finally put this together to obtain a description of the space of neighborhoods up to formal equivalence.

Reçu le : 2018-07-09
Révisé le : 2019-04-02
Accepté le : 2019-07-11
Première publication : 2020-10-01
Classification : 32Q57,  32G12,  37F75
Mots clés: voisinages formels, feuilletages
@unpublished{AIF_0__0_0_A16_0,
     author = {Thom, Olivier},
     title = {Formal classification of two-dimensional neighborhoods of genus $g\ge 2$ curves with trivial normal bundle},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Thom, Olivier. Formal classification of two-dimensional neighborhoods of genus $g\ge 2$ curves with trivial normal bundle. Annales de l'Institut Fourier, à paraître, 22 p.

[1] Arnol’d, Vladimir Igorevich Bifurcations of invariant manifolds of differential equations, and normal forms of neighborhoods of elliptic curves, Funkts. Anal. Prilozh., Volume 10 (1976) no. 4, pp. 1-12 | MR 0431285

[2] Camacho, César; Movasati, Hossein Neighborhoods of analytic varieties, Monografías del Instituto de Matemática y Ciencias Afines, Volume 35, Instituto de Matemática y Ciencias Afines; Pontificia Universidad Católica del Perú, 2003, v+90 pages | MR 2010707 | Zbl 1243.32006

[3] Cerveau, Dominique; Moussu, Robert Groupes d’automorphismes de (C,0) et équations différentielles ydy+=0, Bull. Soc. Math. Fr., Volume 116 (1988) no. 4, pp. 459-488 | MR 1005391

[4] Claudon, Benoît; Loray, Frank; Pereira, Jorge Vitório; Touzet, Frédéric Compact leaves of codimension one holomorphic foliations on projective manifolds, Ann. Sci. Éc. Norm. Supér., Volume 51 (2018) no. 6, pp. 1457-1506 | MR 3940902

[5] Grauert, Hans Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., Volume 146 (1962), pp. 331-368 | Article | MR 0137127

[6] Il’yashenko, Yulij S. Imbeddings of positive type of elliptic curves into complex surfaces, Tr. Mosk. Mat. O.-va, Volume 45 (1982), pp. 37-67 | MR 704627

[7] Loray, Frank Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. (2006) (https://hal.archives-ouvertes.fr/hal-00016434)

[8] Loray, Frank; Thom, Olivier; Touzet, Frédéric Two dimensional neighborhoods of elliptic curves: formal classification and foliations, Mosc. Math. J., Volume 19 (2019) no. 2, pp. 357-392

[9] Mishustin, Mikhail B. Neighborhoods of Riemann curves in complex surfaces, Funkts. Anal. Prilozh., Volume 29 (1995) no. 1, pp. 25-40 | Article | MR 1328536

[10] Mishustin, Mikhail B. On foliations in neighborhoods of elliptic curves, Arnold Math. J., Volume 2 (2016) no. 2, pp. 195-199 | Article | MR 3504349

[11] Neeman, Amnon Ueda theory: theorems and problems, Mem. Am. Math. Soc., Volume 81 (1989) no. 415, p. vi+123 | Article | MR 990364

[12] Thom, Olivier Classification locale de bifeuilletages holomorphes sur les surfaces complexes, Bull. Braz. Math. Soc. (N.S.), Volume 47 (2016) no. 4, pp. 989-1005 | Article | MR 3582024

[13] Thom, Olivier Structures bifeuilletées en codimension 1 (2017) (https://www.theses.fr/2017REN1S064) (Ph. D. Thesis)

[14] Ueda, Tetsuo On the neighborhood of a compact complex curve with topologically trivial normal bundle, J. Math. Kyoto Univ., Volume 22 (1982/83) no. 4, pp. 583-607 | Article | MR 685520