Specialness and Isotriviality for Regular Algebraic Foliations

Amerik, Ekaterina; Campana, Frédéric

Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2923-2950.

Nous montrons l’isotrivialité des feuilles d’un feuilletage $ℱ$ 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 $ℱ$ 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.

We show that an everywhere regular foliation $ℱ$ 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.

Classification : 14C05, 14D06, 14E22, 14E30, 14E40, 14J32
Keywords: algebraic foliations, isotriviality, orbifold divisors, special quasi-projective manifolds
Mot clés : feuilletage algébrique, isotrivialité, diviseurs orbifoldes, varétés quasi-projectives spéciales

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