We show that an everywhere regular foliation on a quasi-projective manifold, such that all of its leaves are compact with semi-ample canonical bundle, has isotrivial family of leaves when the orbifold base of this family is special. The specialness condition means that for any , the -th exterior power of the logarithmic extension of its conormal bundle does not contain any rank-one subsheaf of maximal possible Kodaira dimension . This condition is satisfied, for example, in the very particular case when the Kodaira dimension of the determinant of the logarithmic extension of the conormal bundle vanishes. Motivating examples are given by the “algebraically coisotropic” submanifolds of irreducible hyperkähler projective manifolds.
Nous montrons l’isotrivialité des feuilles d’un feuilletage partout régulier et à feuilles compactes sur une variété quasi-projective lorsque la base orbifolde de la famille des feuilles est spéciale. Cette dernière condition signifie que, pour tout , la puissance extérieure -ième de l’extension logarithmique du fibré conormal de ne contient aucun sous-faisceau de rang un de dimension de Kodaira maximale . Cette condition est satisfaite, par exemple, dans le cas très particulier où la dimension de Kodaira du déterrminant de l’extension logarithmique du fibré conormal est nulle. Des exemples de cette situation sont fournis par les sous-variétés « algébriquement coisotropes » des variétés hyperkählériennes irréductibles projectives.
Keywords: algebraic foliations, isotriviality, orbifold divisors, special quasi-projective manifolds
Mot clés : feuilletage algébrique, isotrivialité, diviseurs orbifoldes, varétés quasi-projectives spéciales
@article{AIF_2018__68_7_2923_0, author = {Amerik, Ekaterina and Campana, Fr\'ed\'eric}, title = {Specialness and {Isotriviality} for {Regular} {Algebraic} {Foliations}}, journal = {Annales de l'Institut Fourier}, pages = {2923--2950}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {7}, year = {2018}, doi = {10.5802/aif.3231}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.3231/} }
TY - JOUR AU - Amerik, Ekaterina AU - Campana, Frédéric TI - Specialness and Isotriviality for Regular Algebraic Foliations JO - Annales de l'Institut Fourier PY - 2018 SP - 2923 EP - 2950 VL - 68 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3231/ DO - 10.5802/aif.3231 LA - en ID - AIF_2018__68_7_2923_0 ER -
%0 Journal Article %A Amerik, Ekaterina %A Campana, Frédéric %T Specialness and Isotriviality for Regular Algebraic Foliations %J Annales de l'Institut Fourier %D 2018 %P 2923-2950 %V 68 %N 7 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3231/ %R 10.5802/aif.3231 %G en %F AIF_2018__68_7_2923_0
Amerik, Ekaterina; Campana, Frédéric. Specialness and Isotriviality for Regular Algebraic Foliations. Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2923-2950. doi : 10.5802/aif.3231. http://archive.numdam.org/articles/10.5802/aif.3231/
[1] Characteristic foliation on non-uniruled smooth divisors on hyperkähler manifolds, J. Lond. Math. Soc., Volume 95 (2017) no. 1, pp. 115-127 | Zbl
[2] Algebraic fibre spaces and curvature of higher direct image sheaves (2017) (https://arxiv.org/abs/1704.02279)
[3] Orbifolds, special varieties and classification theory, Ann. Inst. Fourier, Volume 54 (2004) no. 3, pp. 499-665 | Zbl
[4] Orbifoldes géométriques spéciales et classification biméromorphe des variétés Kählériennes compactes, J. Inst. Math. Jussieu, Volume 10 (2011) no. 4, pp. 809-934 | Zbl
[5] Orbifold generic semi-positivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier, Volume 65 (2015) no. 2, pp. 835-861 | Zbl
[6] Foliations with positive slopes and birational stability of the orbifold cotangent bundles (2017) (https://arxiv.org/abs/1508.02456)
[7] Positivité du fibré cotangent logarithmique et Conjecture de Shafarevich–Viehweg, Séminaire Bourbaki. Volume 2015/2016 (Astérisque), Volume 390, Société Mathématique de France, 2015, pp. 27-63 | Zbl
[8] On the Frobenius integrability of certain holomorphic p-forms, Complex geometry, Springer, 2000, pp. 93-98 | Zbl
[9] Foliations with all leaves compact, Topology, Volume 26 (1977), pp. 13-32 | Zbl
[10] Characteristic foliation on a hypersurface of general type in a projective symplectic manifold, Compos. Math., Volume 146 (2010) no. 2, pp. 497-506 | Zbl
[11] Families over special base manifolds and a conjecture of Campana, Math. Z., Volume 269 (2011) no. 3, pp. 847-878 | Zbl
[12] Positive sheaves of differentials coming from coarse moduli spaces, Ann. Inst. Fourier, Volume 61 (2011) no. 6, pp. 2277-2290 | Zbl
[13] Vector Bundles on Complex Projective Spaces, Progress in Mathematics, 3, Birkhäuser, 1980, vii+389 pages | Zbl
[14] Global stability for holomorphic foliations on Kähler manifolds, Qual. Theory Dyn. Syst., Volume 2 (2001) no. 2, pp. 381-384
[15] Viehweg’s hyperbolicity conjecture for families with maximal variation, Invent. Math., Volume 208 (2017) no. 3, pp. 677-713 | Zbl
[16] The isotriviality of smooth families of canonically polarized manifolds over a special quasi-projective base, Compos. Math., Volume 152 (2016) no. 7, pp. 1421-1434 | Zbl
[17] Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 30, Springer, 1995, viii+320 pages | Zbl
[18] Base spaces of non-isotrivial families of minimal models, Springer, 2002, pp. 279-328 | Zbl
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