Specialness and Isotriviality for Regular Algebraic Foliations
Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2923-2950.

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 p>0, the p-th exterior power of the logarithmic extension of its conormal bundle does not contain any rank-one subsheaf of maximal possible Kodaira dimension p. 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 p>0, la puissance extérieure p-ième de l’extension logarithmique du fibré conormal de ne contient aucun sous-faisceau de rang un de dimension de Kodaira maximale p. 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.

Published online:
DOI: 10.5802/aif.3231
Classification: 14C05,  14D06,  14E22,  14E30,  14E40,  14J32
Keywords: algebraic foliations, isotriviality, orbifold divisors, special quasi-projective manifolds
     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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.3231/}
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.3231/

[1] Amerik, Ekaterina; Campana, Frédéric Characteristic foliation on non-uniruled smooth divisors on hyperkähler manifolds, J. Lond. Math. Soc., Tome 95 (2017) no. 1, pp. 115-127 | Zbl: 06775071

[2] Berndtsson, Bo; Păun, Mihai; Wang, Xu Algebraic fibre spaces and curvature of higher direct image sheaves (2017) (https://arxiv.org/abs/1704.02279)

[3] Campana, Frédéric Orbifolds, special varieties and classification theory, Ann. Inst. Fourier, Tome 54 (2004) no. 3, pp. 499-665 | Zbl: 1062.14014

[4] Campana, Frédéric Orbifoldes géométriques spéciales et classification biméromorphe des variétés Kählériennes compactes, J. Inst. Math. Jussieu, Tome 10 (2011) no. 4, pp. 809-934 | Zbl: 1236.14039

[5] Campana, Frédéric; Păun, Mihai Orbifold generic semi-positivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier, Tome 65 (2015) no. 2, pp. 835-861 | Zbl: 1338.14012

[6] Campana, Frédéric; Păun, Mihai Foliations with positive slopes and birational stability of the orbifold cotangent bundles (2017) (https://arxiv.org/abs/1508.02456)

[7] Claudon, Benoît Positivité du fibré cotangent logarithmique et Conjecture de Shafarevich–Viehweg, Séminaire Bourbaki. Volume 2015/2016 (Astérisque) Tome 390, Société Mathématique de France, 2015, pp. 27-63 | Zbl: 1338.32019

[8] Demailly, Jean-Pierre On the Frobenius integrability of certain holomorphic p-forms, Complex geometry, Springer, 2000, pp. 93-98 | Zbl: 1011.32019

[9] Edwards, Robert; Millett, Kenneth; Sullivan, Dennis Foliations with all leaves compact, Topology, Tome 26 (1977), pp. 13-32 | Zbl: 0356.57022

[10] Hwang, Jun-Muk; Viehweg, Eckart Characteristic foliation on a hypersurface of general type in a projective symplectic manifold, Compos. Math., Tome 146 (2010) no. 2, pp. 497-506 | Zbl: 1208.37031

[11] Jabbusch, Kelly; Kebekus, Stefan Families over special base manifolds and a conjecture of Campana, Math. Z., Tome 269 (2011) no. 3, pp. 847-878 | Zbl: 1238.14024

[12] Jabbusch, Kelly; Kebekus, Stefan Positive sheaves of differentials coming from coarse moduli spaces, Ann. Inst. Fourier, Tome 61 (2011) no. 6, pp. 2277-2290 | Zbl: 1253.14009

[13] Okonek, Christian; Schneider, Michael; Spindler, Heinz Vector Bundles on Complex Projective Spaces, Progress in Mathematics, Tome 3, Birkhäuser, 1980, vii+389 pages | Zbl: 0438.32016

[14] Pereira, Jorge V. Global stability for holomorphic foliations on Kähler manifolds, Qual. Theory Dyn. Syst., Tome 2 (2001) no. 2, pp. 381-384

[15] Popa, Mihnea; Schnell, Christian Viehweg’s hyperbolicity conjecture for families with maximal variation, Invent. Math., Tome 208 (2017) no. 3, pp. 677-713 | Zbl: 1375.14043

[16] Taji, Behrouz The isotriviality of smooth families of canonically polarized manifolds over a special quasi-projective base, Compos. Math., Tome 152 (2016) no. 7, pp. 1421-1434 | Zbl: 06619361

[17] Viehweg, Eckart Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Tome 30, Springer, 1995, viii+320 pages | Zbl: 0844.14004

[18] Viehweg, Eckart; Zuo, Kang Base spaces of non-isotrivial families of minimal models, Springer, 2002, pp. 279-328 | Zbl: 1006.14004

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