Specialness and Isotriviality for Regular Algebraic Foliations

Abstract

Contents 1. Introduction 2. Regular Algebraic Foliations. Compactification. 3. The Log-conormal sheaf of F . 4. Specialness 5. Isotriviality criterion 6. Viehweg-Zuo sheaves 7. Reeb Stability Theorem 8. Elimination of multiple fibres by base-change 9. Orbifold geometry 9.1. Orbifold bases 9.2. Orbifold morphisms 9.3. Smooth orbifold bases of equidimensional fibrations 9.4. Integral parts of orbifold tensors 9.5. Lifting and descent of integral parts of orbifold tensors 9.6. Special smooth orbifolds, proof of the isotriviality criteria. 10. Two examples 10.1. Coisotropic submanifolds 10.2. Boundary of codimension at least 2. 11. Appendix: proof of Theorem 9.15 ReferencesAbstract. We show that an everywhere regular foliation F with compact canonically polarized leaves on a quasi-projective manifold X has isotrivial family of leaves when the orbifold base of this family is special. By [2], the same proof works in the case where the leaves have trivial canonical bundle. The specialness condition means that the p-th exterior power of the logarithmic extension of its conormal bundle does not contain any rank-one subsheaf of maximal Kodaira dimension p, for any p > 0. 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.

On algebraically coisotropic submanifolds of holomorphic symplectic manifolds

“…When K F is semiample, the results of [3] (Theorem 5.1, Corollary 5.2) apply, yielding that f is an isotrivial fibration. Moreover by proposition 12, f has no multiple fiber in codimension one.…”

Orbifold slope-rational connectedness

“…Let now α be a movable class on X (see [14] for this notion, and the ones to follow, which were introduced there). The notions of slope, (semi-)stability, Harder-Narasimhan filtrations of torsion-free sheaves E on X, with respect to any movable class α, are defined exactly as in the classical case of polarisations, with the same properties 1 . In particular, the property µ α,min (E) > 0 means that any quotient Q of E has a strictly positive α-slope.…”