Characteristic foliation on non-uniruled smooth divisors on hyperkähler manifolds

Amerik, Ekaterina; Campana, Frédéric

doi:10.1112/jlms.12008

Cited by 7 publications

(46 citation statements)

References 24 publications

(37 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This answers positively a question raised in [1] for X quasi-projective (instead of quasi-Kähler there). It is quite likely that this more general case can be handled by similar arguments.…”

Section: Isotriviality Criterionsupporting

confidence: 69%

“…The order of the holonomy group along F b , b ∈ B, is also the multiplicity of F b as a scheme-theoretic fibre of f . This fibration is "orbi-smooth" in the sense that all of its scheme-theoretic fibres have smooth reduced support, and B has quotient singularities (see [1] for details). We choose a smooth compactification (X, D) such that D = X − X is a simple normal crossing divisor, in such a way that that this fibration extends to a holomorphic fibration f : X → B with B normal, and such that…”

Section: Regular Algebraic Foliations Compactificationmentioning

confidence: 99%

See 1 more Smart Citation

Specialness and Isotriviality for Regular Algebraic Foliations

Amerik¹,

Campana²

2018

Annales De l'Institut Fourier

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Isotriviality Criterionsupporting

confidence: 69%

Section: Regular Algebraic Foliations Compactificationmentioning

confidence: 99%

Specialness and Isotriviality for Regular Algebraic Foliations

Amerik¹,

Campana²

2018

Annales De l'Institut Fourier

Self Cite

View full text Add to dashboard Cite

show abstract

“…Let X be a projective manifold, and let F ⊂ T X be a holomorphic foliation. Given a movable class α ∈ H n−1,n−1 (X, R) on X, the condition: µ α,min (F ) > 0 means that the inequality of intersection numbers: (1) c 1 (Q).α > 0, holds for any non-zero quotient F → Q → 0. Our first main result is:…”

Section: Introductionmentioning

confidence: 99%

“…Structure of the text. 1 As pointed out by A. Langer, there is a very serious gap in the proof of Theorem 1.4 of [17]. On page 49, the reference [18] is indeed used in a context which is not covered by [18].…”

Section: Introductionmentioning

confidence: 99%

Foliations with positive slopes and birational stability of orbifold cotangent bundles

Campana

Păun

2019

Publ.math.IHES

Self Cite

View full text Add to dashboard Cite

Let X be a smooth connected projective manifold, together with an snc orbifold divisor ∆, such that the pair (X, ∆) is log-canonical. If K X + ∆ is not pseudo-effective, we show, among other things, that any quotient of its orbifold cotangent bundle has a pseudo-effective determinant. This improves considerably our previous result [18], where generic positivity instead of pseudoeffectivity was obtained. One of the new ingredients in the proof is a version of the Bogomolov-McQuillan algebraicity criterion for holomorphic foliations whose minimal slope with respect to a movable class (instead of an ample complete intersection class) is positive.

show abstract

“…A good evidence for Conjecture 0.4 is provided by the results of Charles and Pacienza [9], which deal with the deformations of S [n] (case i = 1), and the deformations of S [2] , (case i = 2), and Lin [16] who constructs constant cycles Lagrangian subvarieties in hyper-Kähler manifolds admitting a Lagrangian fibration. Another evidence is given by the complete family of hyper-Kähler 8-folds constructed by Lehn-Lehn-Sorger-van Straten in [17] that we will study in Section 4 (see Corollary 4.9).…”

Section: Introductionmentioning

confidence: 99%

Remarks And Questions On Coisotropic Subvarieties and 0-Cycles of Hyper-Kähler Varieties

Voisin

2016

Progress in Mathematics

121

View full text Add to dashboard Cite

This paper proposes a conjectural picture for the structure of the Chow ring CH * (X) of a (projective) hyper-Kähler variety X, that seems to emerge from the recent papers [9], [23], [24], [25], with emphasis on the Chow group CH 0 (X) of 0-cycles (in this paper, Chow groups will be taken with Q-coefficients). Our motivation is Beauville's conjecture (see [5]) that for such an X, the Bloch-Beilinson filtration has a natural, multiplicative, splitting. This statement is hard to make precise since the Bloch-Beilinson filtration is not known to exist, but for 0-cycles, this means thatwhere the decomposition is given by the action of self-correspondences Γ i of X, and where the group CH 0 (X) i depends only on (i, 0)-forms on X (the correspondence Γ i should act as 0 on H j,0 for j = i, and Id for i = j). We refer to the paragraph 0.1 at the end of this introduction for the axioms of the Bloch-Beilinson filtration and we will refer to it in Section 3 when providing some evidence for our conjectures. Note that a hyper-Kähler variety X has H i,0 (X) = 0 for odd i, so the Bloch-Beilinson filtration F BB has to satisfy F i BB CH 0 (X) = F i+1 BB CH 0 (X) when i is odd. Hence we are only interested in the F 2i BBlevels, which we denote by F ′ i BB . Note also that there are concrete consequences of the Beauville conjecture that can be attacked directly, namely, the 0-th piece CH(X) 0 should map isomorphically via the cycle class map to its image in H * (X, Q) which should be the subalgebra H * (X, Q) alg of algebraic cycle classes, since the Bloch-Beilinson filtration is conjectured to have F 1 BB CH * (X) = CH * (X) hom . Hence there should be a subalgebra of CH * (X, Q) which is isomorphic to the subalgebra H 2 * (X, Q) alg ⊂ H 2 * (X, Q) of algebraic classes. Furthermore, this subalgebra has to contain NS(X) = Pic(X). Thus a concrete subconjecture is the following prediction (cf. This conjecture has been enlarged in [28] to include the Chow-theoretic Chern classes of X, c i (X) := c i (T X ) which should thus be thought as being contained in the 0-th piece of the conjectural Beauville decomposition. Our purpose in this paper is to introduce a new set of classes which should also be put in this 0-th piece, for example, the constant cycles subvarieties of maximal dimension (namely n, with dim X = 2n, because they have to be isotropic, see Section 1) and their higher dimensional generalization, which are algebraically coisotropic. Let us explain the motivation for this, starting from the study of 0-cycles.Based on the case of S [n] where we have the results of [4], [22], [25] that concern the CH 0 group of a K3 surface but will be reinterpreted in a slightly different form in Section 2, we introduce the following decreasing filtration S · on CH 0 (X) for any hyper-Kähler manifold *

show abstract

Characteristic foliation on non-uniruled smooth divisors on hyperkähler manifolds

Cited by 7 publications

References 24 publications

Specialness and Isotriviality for Regular Algebraic Foliations

Specialness and Isotriviality for Regular Algebraic Foliations

Foliations with positive slopes and birational stability of orbifold cotangent bundles

Remarks And Questions On Coisotropic Subvarieties and 0-Cycles of Hyper-Kähler Varieties

Contact Info

Product

Resources

About