Characteristic foliations — A survey

This is a survey article, with essentially complete proofs, of a series of recent results concerning the geometry of the characteristic foliation on smooth divisors in compact hyperkähler manifolds, starting with work by Hwang–Viehweg, but also covering articles by Amerik–Campana and Abugaliev. The restriction of the holomorphic symplectic form on a hyperkähler manifold X$X$ to a smooth hypersurface D⊂X$D\subset X$ leads to a regular foliation F⊂TD${\mathcal {F}}\subset {\mathcal {T}}_D$ of rank 1, the characteristic foliation. The picture is complete in dimension 4 and shows that the behaviour of the leaves of F${\mathcal {F}}$ on D$D$ is determined by the Beauville–Bogomolov square q(D)$q(D)$ of D$D$ . In higher dimensions, some of the results depend on the abundance conjecture for D$D$ .


Main theorem and motivation
Throughout, D Ă X denotes a smooth connected hypersurface in a compact hyperkähler manifold X of complex dimension 2n, i.e.X is a simply connected, compact Kähler manifold such that H 0 pX, Ω 2 X q is spanned by a holomorphic symplectic form σ. Usually X will be in addition assumed to be projective, although one expects all results to hold in general.
The symplectic form σ induces a regular foliation of rank one on D, i.e. a line sub-bundle F Ă T D .We shall denote a generic leaf of the foliation by L and its Zariski closure by L. The space of leaves will be denoted D{F.These notions will all be recalled in Sections 2 and 3.
We will discuss the following table.The ultimate goal, only partially met at the moment, would be to establish the equivalence of all assertions in each row.We will throughout assume n ą 1, but see Remark 1.1.

of general type
This review was prepared in the context of the seminar organised by the ERC Synergy Grant HyperK, Grant agreement ID 854361.The talk was delivered on December 10, 2021.
The table was originally proposed by Campana, cf. the work of Amerik-Guseva cf.[AG16].Essentially, in each row the four conditions are known or at least expected to be equivalent to each other.The assumption on D to be smooth is essential, see Section 8.2.The following serves as a guide for what will be discussed in the subsequent sections, where precise references will be provided.The implication piiiq + 3 pivq is currently only proved assuming the abundance conjecture for D, so the proof is only complete for n " 2, cf.[AG16].
Additionally, we will also provide a direct argument for pivq § 6.5 + 3 piq .
Remark 1.1.Let us consider the case n " 1, i.e.X a K3 surface.Then, a smooth hypersurface D Ă X is just a smooth curve.Clearly, in this case (i) holds in all three Cases (1), (2), and (3).The equivalence of the other conditions (ii), (iii), and (iv) in (1) and (2) and of (iii) and (iv) in (3) is well known.
We do not claim any originality for the results presented in this survey, although we sometimes give alternative arguments or provide more details.We hope that the survey contributes to the dissemination of these results.

Preparations I: Linear algebra of the characteristic foliation
We collect some linear algebra results and discuss applications to the geometry of the leaves of a foliation.
2.1.We begin with discussing some easy linear algebra results that will be used throughout the later sections.
Let W be a vector space together with a symplectic structure σ, i.e. σ P Ź 2 W ˚such that the induced map σ : W " / / W ˚is an isomorphism.In this situation, the dimension of W is even, so dim W " 2n.
Lemma 2.1.Assume V Ă W is a subspace of codimension one.Then the subspace Similarly, if U Ă W is of codimension two, then either dim kerpσ| U : U / / U ˚q " 2 or σ| U P Ź 2 U ˚is non-degenerate, i.e. kerpσ| U q " 0.
Proof.Since W " / / W ˚/ / / / V has a one-dimensional kernel, we have dim F ď 1.Furthermore, since dim V " 2n ´1 is odd, the alternating form σ| V P Ź 2 V ˚cannot be non-degenerate, i.e. kerpσ| V q ‰ 0. Hence, dim F " 1.The proof of the second assertion is analogous.
Lemma 2.2.Assume V Ă W is of codimension one and let F " kerpσ| V q Ă V .Then σ naturally induces a symplectic structure σ on V {F .

Proof. By definition of F , the restriction σ|
Furthermore, since σ is alternating, σ| V takes values in pV {F q ˚Ă V ˚.Hence, for dimension reasons, σ : Here are a few more concepts from linear algebra: A subspace U Ă W of codimension c of a symplectic vector space pW, σq is called isotropic if σ| U P Ź 2 U ˚is trivial or, equivalently, if Proof.The first claim follows from the commutativity of the diagram and the assumption that U is Lagrangian, which implies that U " kerpW " / / W ˚/ / / / U ˚q.
For the second assertion apply Lemma 2.1.Since F Ă U Ă V , the restriction σ| U is degenerate and, therefore, dim kerpσ U q " 2.

2.2.
A regular foliation of a smooth variety (or complex manifold) D is a locally free subsheaf F Ă T D with a locally free quotient and such that F is integrable, i.e. rF, Fs Ă F. Note that the integrability condition is automatically satisfied if rkpFq " 1, which is the case of interest to us.
A leaf of a foliation is a maximal connected and immersed complex submanifold L Ă D (more precisely, a complex manifold together with an injective immersion into D) with F| L " T L as subsheaves of T D | L .The integrability condition ensures that there exists a (unique) leaf through any point of D. This is the Frobenius integrability theorem, cf.[Man97, §5, Thm.2].A submanifold Z Ă D is invariant under the foliation if F| Z Ă T Z as subsheaves of T D | Z .If Z is a singular subvariety of D, then we call Z invariant if its smooth locus is invariant.It is not hard to see that the Zariski closure of an invariant complex submanifold is invariant.Also note that every leaf L intersecting an invariant submanifold Z Ă D is contained in its closure.
A leaf L Ă D is typically not closed.Its Zariski closure L Ă D can be identified with the smallest subvariety containing L that is invariant under the foliation.
Consider now the case of a smooth hypersurface D Ă X of a compact hyperkähler manifold.By virtue of Lemma 2.1, the kernel is a sub-line bundle with locally free kernel.It is called the characteristic foliation of the hypersurface D Ă X and was first studied by Hwang and Viehweg [HV10].
Lemma 2.4.The normal bundle of the characteristic foliation N F :" T D {F is naturally endowed with a symplectic structure and F » ω D.
In particular, any local transverse section Σ of a leaf L Ă D has a natural symplectic structure.
Proof.The first assertion follows from Lemma 2.2.The existence of a symplectic structure on N F implies detpN F q » O D and hence F » detpT D q » ω D.
Remark 2.5.For foliations in general detpFq ˚is often called the canonical bundle ω F of the foliation.For the characteristic foliation we thus have ω F » ω D .Note that for n " 1, this becomes F » T D which is not interesting so that we usually assume n ą 1.Then, according to [Dr17, Thm.1.1], detpFq » ω D cannot be big and nef.
The geometric versions of isotropic, coisotropic, and Lagrangian for subspaces of a symplectic vector space are readily defined: For example, a subvariety Z Ă X is coisotropic if the rank of σ| Z : T Z / / Ω Z (say over the smooth locus of Z) is 2 dimpZq ´dimpXq or, equivalently, if rkpkerpσ| Z qq " codimpZ Ă Xq.
The geometric analogue of Lemma 2.3 is the following.
Corollary 2.6.Assume D Ă X is a smooth hypersurface of a compact hyperkähler manifold.
(i) If T Ă X is a smooth Lagrangian submanifold that is contained in D, then T is covered by leaves or, equivalently, every leaf L Ă D intersecting T is contained in T .
(ii) Furthermore, any invariant subvariety Z Ă X of codimension two that is contained in D Ă X is coisotropic.

Preparations II: Space of leaves
There is no standard text on foliations on complex manifolds or algebraic varieties, but see e.g.[CN85].The arguments typically rely very much on the differentiable theory.The holomorphic version of Reeb's classical stability theorem, cf.[HV10, KCT07], is one example.We recommend [AC14, Sec.2.1] for further comments.
3.1.Consider a foliation F (of rank one) on a compact complex manifold D. The space of leaves is the quotient D{F by the equivalence relation that identifies two points if they are contained in the same leaf.The quotient topology is often complicated and frequently non-Hausdorff, but the projection π : D / / D{F is open, i.e. for any open set U Ă D the union of all leaves intersecting U (its saturation) is again an open subset.For more information see [CN85, Ch.III].A typical example is that of a P 1 -bundle π : D " PpEq / / Z with F " T π .In this situation, D{F " Z.We will come back to the local structure of D{F and the map π : D / / D{F in the case that the foliation is algebraically integrable, i.e. when every leaf is compact. 1 3.2.Let L " L Ă D be a compact leaf.For a fixed point x P L we pick a small transversal section x P Σ x Ă D (think of it as a germ of a transversal section).Consider a closed loop γ : r0, 1s / / L with γp0q " γp1q " x and pick a point y P Σ x close to x.Then there exists a differentiable map Φ : Σ x ˆr0, 1s / / X such that Φpy, 0q " Φpy, 1q, Φp0, tq " γptq, and Φpy, 0q " y.The pull-back of the foliation F defines a real foliation of rank one on Σ x ˆS1 .Starting with a point py, 0q and integrating defines a path γ y : r0, 1s / / Σ x ˆS1 satisfying γ y ptq " pρ γ,y ptq, tq and γ y p0q " py, 0q.Note, however, that this path is not necessarily closed, so possibly γ y p1q ‰ py, 0q.
It turns out that the map y ✤ / / ρ γ,y p1q only depends on the homotopy class of γ, which gives rise to the following.
The leaf has finite holonomy if the image of ρ, the holonomy group Note that the image of ρ only depends on x up to conjugation.In particular, the property of a leaf to have finite holonomy does not depend on the chosen base point.
Since we are interested in foliations of rank one, a compact leaf will be a compact complex curve.If this curve is rational, i.e.L " L » P 1 , then it has automatically finite (and in fact trivial) holonomy.Also note that due to a result of Holmann [Ho80,Prop. 4.2], one knows that if all leaves of a foliation on a Kähler manifold are compact, then they all have finite holonomy. 2Theorem 3.2.Assume that a foliation F of rank one on a smooth projective variety D has one leaf isomorphic to P 1 .Then F is algebraically integrable and all leaves are curves isomorphic to P 1 .
Proof.According to a result of Pereira [Pe01, Thm.1], for a foliation on a compact Kähler manifold the existence of one compact leaf with finite holonomy implies that all leaves are compact with finite holonomy.This proves that F is algebraically integrable.Reeb stability [HV10, Prop.2.5] then implies that all leaves are isomorphic to each other. 32 In fact, Holman [Ho80, Prop.4.2] proved that for a holomorphic foliation with only compact leaves stability is equivalent to finite holonomy.Earlier results in this direction are due to Epstein [Ep76].
3 A word on the name 'Reeb stability': The foliation is stable if all leaves are stable.Reeb stability in the holomorphic context as in [HV10, KCT07] can be viewed as saying that compact leaves with finite holonomy are stable.

3.3.
Let us come back to the space of leaves D{F and the map π : D / / D{F for an algebraically integrable foliation (of rank one) on a smooth projective variety D. We collect the facts that will be used at various places later: 4 ‚ The map π : x ✤ / / |G Lx | ¨rL x s defines a holomorphic map from D to the Chow variety (or Barlet space), cf.[HV10, Prop.2.5] of [MM03].Here, L x is the unique leaf through x and G Lx is its holonomy group.
‚ The quotient D{F can be identified with (the normalization of) the image of π.In particular, D{F is an algebraic variety and π : D / / D{F is a proper morphism.The map is in general not flat.Indeed, by 'miracle flatness', the flatness of the equidimensional morphism π is equivalent to the smoothness of the leaf space D{F, which is discussed next.
‚ Assume x P Σ x Ă D is a transversal section of a leaf x P L as in Section 3.2.Then locally Σ x {G x is a chart of D{F at the point corresponding to the leaf L x , cf. [HV10, Thm.2.4].In particular, D{F has quotient singularities.

‚ Assume the fibres of D
/ / D{F are rational curves.Then the description of local charts shows that D{F is a smooth projective variety, for in this case π 1 pL x q " t1u and hence G Lx " t1u.In the differentiable setting this is [He60] and for a discussion in our setting see e.g.[Sa09,Lem. 5].If, furthermore, F is the characteristic foliation of a smooth hypersurface in a projective hyperkähler manifold, then D{F comes with a natural symplectic structure.
‚ For a dense open subset of points x P D the leaf L x through x has trivial holonomy ‚ The scheme-theoretic fibre of π : D / / D{F over a point rLs P D{F corresponding to a leaf with trivial holonomy |G L | " 1 is the leaf L. The fibre is non-reduced over points with non-trivial holonomy; more precisely, it is a multiple fibre with multiplicity |G L | ‰ 1.

3.4.
It is easy to prove that a smooth curve C Ă S in a K3 surface with pC.Cq ě 0 is nef.The following is the hyperkähler analogue of this fact. 5Proposition 3.3.Let D Ă X be a smooth hypersurface in a projective hyperkähler manifold X.If qpDq ě 0, then D is nef.
Proof.Assume D is not nef.Then there exists an irreducible curve C Ă X with D ¨C ă 0. The latter implies C Ă D and degpω D | C q ă 0, which by Lemma 2.4 shows degpF| C q ą 0, i.e.F| C is ample.
However, the latter implies that the foliation F is algebraic, i.e. all leaves are compact or, equivalently, algebraic curves.This is a consequence of a much more general result that was 4 These results seem well known to the experts but we could not find a source with complete proofs.Thanks to J.-B.Bost for an instructive email exchange. 5We wish to thank R. Abugaliev for communicating this result to us.It seems known to some experts, but has not been written down anywhere.
original proved by  and Bost [Bo01] with details provided by Kebekus, Solá Conde, and Toma [KCT07, Thm. 1 & 2]: If the restriction of a foliation to some complete curve C is an ample vector bundle, then the leaf through any point of C is algebraic.Moreover, the leaf through a general point of C is rationally connected and, in fact, all leaves are rationally connected.In fact, according to Theorem 3.2, all we need is one compact rational leaf.6In our situation the result means that all leaves of F are smooth rational curves and, in particular, D is uniruled.Then, the discussion in Section 4.2, which is independent of this proposition, leads to the contradiction qpDq ă 0.

Case (1): Closed leaves
We are proving the equivalence of the conditions (i)-(iv) in Case (1).All results are unconditional.The main reference for this section is [AC14] with priori work [Bo04,Hu03].

(i) ñ (iv):
We assume that the leaves of the foliation have one-dimensional closures and want to show that D is then uniruled. 7irst of all, since the boundary LzL is invariant, it is a union of leaves.However, under our assumption all leaves are one-dimensional and, therefore, all leaves are in fact closed L " L. Thus, all leaves are algebraic curves, i.e.F is algebraically integrable, and the natural projection is a proper morphism between algebraic varieties.
The proof proceeds in three steps.
Step 1: Prove that π has no multiple fibres in codimension one and that the canonical divisor of D{F is trivial.
Step 2: Deduce the isotriviality of π, combining results of Miyaoka and Hwang-Viehweg, and consider a finite quasi-étale cover of D that splits into a product.
Step 3: Reach a contradiction by considering the numerical dimension of ω D .
Step 1. Intuitively, the morphism π : D / / D{F induced by the algebraically integrable foliation F contracts all curves with tangent space contained in the kernel of σ| D .Therefore σ| D should descend to a non-degenerate two-form on the a priori singular space D{F and so we expect (the smooth part of) D{F to have trivial canonical bundle.
For the open set covered by leaves with trivial holonomy this can be made rigorous by taking a locally transverse section Σ to each leaf of the foliation, which can be taken as a local model for D{F near the point corresponding to the leaf.Then, by Lemma 2.4, σ| Σ is symplectic.These symplectic forms glue to a global symplectic form on the open subset of D{F corresponding to leaves with trivial holonomy, see [Sa09,Lem. 6] for some more details.
Looking at the local behavior of the symplectic form around the multiple fibres, Amerik and Campana [AC14] proved the following: Lemma 4.1 (Amerik-Campana).The map π has no multiple fibres in codimension one.Moreover some multiple of the canonical bundle of D{F is trivial.
Proof.Suppose by contradiction that there exists a divisor V Ă D{F such that the fibres over V are multiple of order m ą 1.The statement is local around a generic point 0 P V .We can take a local multisection W over 0 that meets transversally the non-reduced fibres.We choose coordinates pz, tq around 0 P V such that z are coordinates for V and t parametrizes the normal direction.Thus, we can choose coordinates pu, s, wq around W in such a way that W is given by the equation w " 0 and πpu, s, wq " pu, s m q.
By the discussion above, σ n´1 " π ˚α for some form of top degree on the base at least over the complement of V .Locally, α " Gpz, tq ¨dz ^dt, where dz is an n ´2 form, and then |Gpz, tq| " e gpz,tq ¨|t| ´c for some real-valued bounded function g.We claim that c " 1 ´1{m.Assuming for the moment that this is true, then the meromorphic function Gpz, tq has poles of order strictly less than one around t " 0, which is absurd.
To prove the claim we denote by π 0 the restriction of π : D / / D{F to W .In coordinates π 0 pu, s, wq " pu, s m q.By the base change formula, we see that the restriction of σ n´1 to W is σ n´1 | W " π 0 α " Gpu, s m q ¨m ¨sm´1 ¨du ^ds " hpu, sq ¨du ^ds for some function hpu, sq that does not vanish when s " 0. Thus, we can write |Gpz, tq| " |Gpu, s m q| " |hpu, sq| m |s| 1´m " e gpu,sq |s| 1´m " e gpz,tq |t| ´1`1{m which proves the claim.
The singular fibres of π : D / / D{F are simply multiples of smooth curves.By the above lemma we can assume π is smooth over the complement D o {F Ă D{F of a closed set of codimension two and we may assume that D o {F is smooth.Denote by π o : D o / / D o {F the restriction of π.If one leaf is rational, then by Reeb stability, see Theorem 3.2, all the leaves are rational curves and we are done. 8So we can assume all the leaves have positive genus and singular, i.e. multiple, fibres appear in codimension at least two.
Step 2. We want to prove that π o : D o / / D o {F is isotrivial.There are two possibilities depending on the genus g of the general leaf: If g " 1, then the fibration has to be isotrivial, 8 One can avoid using Reeb stability here: Instead of showing (i) ñ (iv) one shows (i) ñ (iii) and then uses Section 4.3 to complete by (iii) ñ (iv).Indeed, qpDq ă 0 is equivalent to ş D c1pFqH 2n´2 ą 0 for some ample divisor H on D. The latter follows if one can show ş D c1pFqπ ˚H 2n´2 0 ą 0 for some ample divisor H0 on D{F, which in turn would follow from ş L c1pFq|L ą 0, i.e. gpLq " 0.
for otherwise one of the fibres of π would be rational, in which case we are done already, cf.Theorem 3.2.The isotriviality is less trivial for g ą 1.It follows from the observation that the following results of Miyaoka-Mori and Hwang-Viehweg contradict each other.Once isotriviality for g " 1 and g ą 1 has been established, one can use the fact that the moduli space M g pN q of curves with a level N -structures, N ě 3, is fine to show that there exists a finite étale cover ∆ / / D o {F such that pull-back D :" D o ˆDo {F ∆ splits as D » ∆ ˆC, where C is the generic fibre of π.Indeed, there exists a finite étale cover ∆ / / D o {F such that the finite cohomology groups H 1 p Dt , Z{N Zq, t P ∆, of the fibres of the pull-back family πo : D :" D o ˆDo {F ∆ / / ∆ form a trivial local system.The induced morphism ∆ / / M g pN q has the property that the pull-back of the universal curve over M g pN q is isomorphic to πo .However, the isotriviality implies that ∆ / / M g pN q is constant and, therefore, D splits as claimed.
Step 3. Since D o {F has trivial canonical bundle, the same holds for ∆.Hence, νp D, ω D q " κp Dq " As the numerical (and also the Kodaira) dimension is preserved under étale maps, one finds Since ω D " O X pDq| D , we have νpD, ω D q " νpX, Dq ´1.However, the numerical dimension of a nef divisor in a hyperkähler manifold can be 0, n or 2n.Since n ą 1, the only possibility is that n " 2 and g ą 1, which is excluded as follows: A fibre S of the canonical map is equivalent as a cycle, up to a multiple, to D ¨D.This means that S is Lagrangian, for ş S σσ " qpDq " 0. Hence, by Corollary 2.6, the leaves of the characteristic foliation must be contained in S and induce a fibration on S of curves of genus at least two.This contradicts the fact that the canonical bundle of S is trivial.
Suppose qpDq ą 0. Then D is contained in the interior of the positive cone and, therefore, also in the interior of the pseudo-effective cone.Hence, D is big [La04, Lem.2.2.3], i.e. h 0 pX, OpkDqq " k 2n , which implies h 0 pD, ω k D q " k 2n´1 contradicting the assumption that D is uniruled.
Suppose qpDq " 0. If D is nef, then ω D is nef too, which again would contradict the assumption that D uniruled.If D is not contained in the closure of the movable cone, then, by Boucksom's duality of movable and pseudo-effective cone [Bo04], it is contained in the interior of the pseudo-effective cone and one argues as above.If D is contained in the boundary of the movable cone, D is the limit of movable divisors and hence its restriction to D is still a limit of effective divisors.However, this implies that ω D » OpDq| D is pseudo-effective which contradicts D uniruled.9 The discussion should be compared to the result [LPT18, Thm.3.7] asserting in the general setting that D is uniruled if and only if ω F is not pseudo-effective.The above discussion can be interpreted as saying that any smooth divisor D Ă X with qpDq ě 0 has a pseudo-effective ω D , thus D cannot be uniruled.

(iii) ñ (iv):
We assume qpDq ă 0 and want to show that D is then uniruled.(The smoothness of D is not essential.)We offer two proofs.
First, it is known that prime exceptional divisors are uniruled [Hu03, Prop.5.4] or [Bo04, Prop.4.7 & Thm.4.3].Indeed, since the positive cone is self-dual, it contains a class α such that qpα, Dq ă 0. Hence, there exists a bimeromorphic map between hyperkähler manifolds f : X X 1 such that f ˚α " ω 1 `ř a i D 1 i for some uniruled divisors D 1 i , positive real numbers a i , and a Kähler class ω 1 , cf. [Hu03] or [Bo04,Thm. 4.3 (ii)].Since the quadratic form is preserved by f , we have 0 ą qpα, Dq " qpω 1 `ř a i D 1 i , f ˚Dq ą ř a i qpD 1 i , f ˚Dq and hence for some i we have qpD 1 i , f ˚Dq ă 0. This implies that f ˚D and D i coincide and that in particular D is uniruled since its push-forward in X 1 is.
Here is a more direct proof relying on the criterion for uniruledness by Miyaoka and Mori [MM86,Miy87] ş X rDs 2 ¨H2n´2 ă 0 for any ample divisor H on X, it implies the result.

(iv) ñ (i):
We assume that D is uniruled and want to prove that the leaves are closed.
By assumption, there exists a dominant map ϕ : P 1 ˆV / / D with dimpV q " 2n´2.Since P 1 admits no non-trivial forms of degree one or two, the pull-back of σ to P 1 ˆV is the pull-back of a holomorphic form on V .This readily shows that all ϕ t pP 1 q Ă D are invariant with respect to the foliation.Hence, the generic leaf is of this form, which proves the claim.See [Dr11,Prop. 4.5] for a generalization to singular uniruled divisors.4.5.(i) ô (ii): Clearly, (ii) implies (i).The converse is part of the discussion in Section 4.1.

Case (2): Lagrangian fibrations
The equivalence of the conditions (i)-(iv) in Case (2) is shown in dimension four.In higher dimensions, the proof assumes that abundance holds for D. The direction (iv) ñ (i) is due to Abugaliev and is the main result of [Ab19].
According to Section 4, qpDq ă 0 implies dim L " 1.To exclude qpDq ą 0, use that according to Section 6.4 10 it would imply that the leaves are dense.

(iii) ñ (iv):
We assume qpDq " 0.Then, by Proposition 3.3, D and hence ω D » OpDq| D are nef.Assuming the abundance conjecture for D, we know that ω D is semi-ample.Hence, by [DHP13, Cor.1.8] also D is semi-ample, 11 i.e. some power OpkDq defines a Lagrangian fibration f : X / / B and, therefore, kD is the pull-back of a divisor in B. Hence, D is vertical.

(iv) ñ (iii):
We assume now that there exists a Lagrangian fibration X / / B such that D is the pre-image (as a set) of a hypersurface H Ă B. Then rDs and f ˚rH s are proportional.Therefore, since rHs n`1 " 0, also rDs n`1 " 0 and hence qpDq " 0.

(iv) ñ (i):
We assume that X comes with a Lagrangian fibration f : X / / B such that D " f ´1pH q (as sets) for some hypersurface H Ă B and want to show that the closure of the generic leaf is of dimension n (and in fact a torus).
Assume first that H is contained in the discriminant locus ∆ Ă B. Then D is algebraically integrable by a result of Hwang and Oguiso [HO09, Thm.1.2].By the results of Section 4, the latter implies that D is uniruled and hence qpDq ă 0, which contradicts (iii) that we proved already in Section 5.3. 12Assume now that H is not contained in the discriminant divisor.Then, since D is smooth, the generic fibre of f | D : D / / H " f pDq is a smooth Lagrangian torus.By Corollary 2.6, the generic leaf is contained in a fibre of D / / f pDq.We have to show that it is dense in there.
10 We leave it to the reader to check that the argument is not circular.
11 This is a highly non-trivial statement asserting that H 0 pX, OpkDqq / / H 0 pD, OpkDq|Dq is surjective for sufficiently divisible k.For an alternative, algebraic argument see [AC14, Cor.5.2]. 12The argument shows that for any component of the discriminant divisor H Ă B the reduction of f ´1pH q cannot be smooth.Either it consists of more than one component, with possibly each component individually smooth, or it is irreducible but singular.
Note that for n " 2 the result is immediate.Indeed, if the generic leaf is not dense in the fibre, then by Section 4 the foliation is algebraically integrable and the leaves are rational curves, which however do not exist in a torus.
Let T :" f ´1ptq, t P f pDq, be a generic fibre.The foliation F Ă T D induces a foliation F| T Ă T T of the abelian variety T .It is well known that the closure of a leaf of a foliation on an abelian variety is a translate of an abelian subvariety.Indeed, observing O F » F| T Ă T T » O 'n F and writing T " C n {Γ, one finds that the leaves of the foliation F are given by the images under the natural projection C n / / / / T of the translates of the line C Ă C n corresponding to F| T Ă T T .The closure of the leaf through the origin then corresponds to the smallest linear subspace C m Ă C n containing the given line and such that Γ X C m Ă C m is a lattice.
Thus, if the abelian variety T is known to be simple, which is frequently the case, then the assertion is immediate.
If T is not simple, then Abugaliev proceeds in two steps.The first is a result of general interest [Ab19, Thm.0.5]. 13  Lemma 5.2 (Abugaliev).Let f : X / / B be a Lagrangian fibration of a projective hyperkähler manifold and let H Ă B be a very ample hypersurface not contained in the discriminant divisor of f .If D " f ´1pH q Ă X is smooth, then for the generic fibre T " f ´1ptq, t P H: Im ˆH˚p X, Qq res X,T / / H ˚pT , Qq ˙" Im ˆH˚p D, Qq res D,T / / H ˚pT , Qq ˙.
Note that the left hand side is known to be isomorphic to H ˚pP n , Qq according to results by Matsushita, Oguiso, Voisin, and Shen-Yin, see the survey [HM21, Thm.2.1] for references.In particular, it is of dimension one in each even degree.
Proof.The assertion is invariant under small deformations of H, which preserve the smoothness of D. One may assume that the intersection H X ∆ with the discriminant locus is sufficiently generic such that π 1 pHz∆q / / / / π 1 pBz∆q is surjective (and in fact an isomorphism for n ą 2) by [De81, Lem.1.4] applied to Bz∆.In particular, the monodromy invariant parts of H ˚pT , Qq for the two families X / / B and D / / H coincide.Thus, Deligne's invariant cycle theorem implies Impres X,T q " H ˚pT , Qq π 1 pBz∆q " H ˚pT , Qq π 1 pHz∆q " Impres D,T q, which concludes the proof.
The idea of the second step is that the family of tori obtained as closures of leaves L Ă L Ă T contained in a fixed generic fibre T is distinguished and hence invariant under monodromy.This gives a cohomology class in H 2k pT, Qq that is invariant under monodromy of the family D / / H. 13 The reader will observe that the result actually holds without assuming that X is hyperkähler or that f is a Lagrangian fibration.
However, classes that are invariant under the monodromy of the family X / / B are all powers of the polarization and, therefore, cannot be realized by proper subtori.5.5.(iv) ñ (ii) ñ (i): Assume first (iv) holds.By Corollary 2.6, the generic leaf L is contained in a fibre of D / / f pDq Ă B. If we allow ourselves to use (iv) ñ (i) in Section 5.4, then the closure L is the generic fibre which is a torus.The second implication (ii) ñ (i) is clear.5.6.Let T " f ´1ptq Ă X be a smooth fibre of a Lagrangian fibration f : X / / B. Then T is isomorphic to an abelian variety and picking a point x P X allows one to write T as a torus T " T x T {Γ.For a hypersurface t P H Ă B, which we assume to be smooth at t, we let D :" f ´1pH q be its pre-image.Since the symplectic structure of X provides an isomorphism T x T » T t B, the tangent space T t H Ă T t B, viewed as a line in T t B, corresponds to a line ℓ H Ă T x T .The image of this line under T x T / / T gives the leaf through x P D of the characteristic foliation on D. If H is chosen very general, then the line ℓ H Ă T x T is very general and, therefore, its image in the quotient T " T x T {Γ is dense.This proves the following.
Proposition 5.3.For a fixed smooth fibre T " f ´1ptq Ă X of a Lagrangian fibration f : X / / B and a very general smooth hypersurface t P H Ă B the leaf of the characteristic foliation of D " f ´1pH q through a point x P T is dense in the fibre T .

Case (3): Dense leaves
Again, the equivalence of the conditions (i)-(iv) holds in dimension four, but assumes that the abundance conjecture holds for D. Hwang and Viehweg [HV10] showed that if D is of general type, the foliation is not algebraically integrable.In the converse direction, in dimension four, Amerik and Campana [AC14] proved that the foliation is algebraically integrable if and only if D is uniruled.The assertion that D being nef and big implies density of the leaves is due to Abugaliev and the main result of [Ab21].6.1.(i) ñ (iii): We assume that dim L " 2n ´1 and want to exclude that qpDq ă 0 or qpDq " 0.
First, by the results of Section 4, we know that the three conditions qpDq ă 0, D uniruled, and dim L " 1 are all equivalent.Hence, qpDq ă 0 is excluded for dim L ą 1.
Next suppose qpDq " 0.Then, by Proposition 3.3, D is nef and hence also ω D » OpDq| D is.Assuming the abundance conjecture for D, we conclude that ω D and, therefore, OpDq are semiample, cf. the argument in Section 5.2.Hence, X comes with a Lagrangian fibration X / / B such that some multiple of D is the pre-image of a divisor in B. However, then the generic leaf is not dense, as a leaf passing through a smooth fibre stays in this fibre, cf. the discussion in Section 5.4.6.2.(iii) ñ (iv): We assume qpDq ą 0. Clearly, if D is ample, then by adjunction ω D » OpDq| D is ample as well and, therefore, D is of general type.If D is only nef, then a priori also ω D is only nef.However, qpDq ą 0 implies ş D c 1 pω D q 2n´1 ą 0, i.e. ω D is big and nef.By the Kawamata-Viehweg vanishing theorem H i pD, ω k D q " 0 for k ą 1 and i ą 0 and by the Hirzebruch-Riemann-Roch formula h 0 pD, ω k D q " k 2n´1 .Hence, D is of general type.Since by Proposition 3.3, any smooth hypersurface D Ă X with qpDq ě 0 is nef, this concludes the proof.
Here is an alternative argument not relying on Proposition 3.3: Since D is contained in the interior of the positive cone, it is also contained in the interior of the pseudo-effective cone and, therefore, big by [La04, Prop.2.2.6], i.e. h 0 pX, OpkDqq " k 2n .Using ω D » OpDq| D eventually shows that ω D is big, i.e. that D is of general type.6.3.(iv) ñ (iii): We assume that D is of general type and want to prove qpDq ą 0 by excluding the other two possibilities qpDq ă 0 and qpDq " 0.
Suppose qpDq ă 0. Then by virtue of Section 4.3, e.g. by applying the Miyaoka-Mori numerical criterion for uniruledness [MM86,Miy87], we know that D is uniruled, so in particular not of general type.
Next, suppose that qpDq " 0, which implies ş D c 1 pω D q 2n´1 " pDq 2n " 0. Now use again Proposition 3.3 to conclude that OpDq and hence ω D » OpDq| D are nef.However, a nef divisor E on a projective variety Z of dimension m is big, i.e. h 0 pZ, OpkEqq " k m , if and only if pEq m ą 0, see [La04, Thm.2.2.16].Since D is assumed to be of general type and so h 0 pD, OpkDq| D q " k 2n´1 , this is a contradiction.6.4.(iii) ñ (i): We assume qpDq ą 0 and want to show that the leaves are dense.We have seen already that qpDq ą 0 implies that D is big and nef.The first step is to prove a version of the Lefschetz hyperplane theorem [Ab21, Prop.3.1].Lemma 6.1.Assume D Ă X is a smooth hypersurface of a hyperkähler manifold that is big and nef.Then the restriction is an isomorphism for i ă dimpDq " 2n ´1.
Proof.For an ample hypersurface this is the content of the Lefschetz hyperplane theorem [La04, Thm.3.1.17].If D is just big and nef, Kawamata-Viehweg vanishing still shows that all higher cohomology groups H i pX, OpkDqq, i ą 0, are trivial.Hence, D deforms sideways with X in any family X / / ∆ for which the line bundle OpDq deforms.However, the very general fibre X t of the universal such deformation has Picard number one.Therefore, a generic deformation D t Ă X t of D Ă X is ample [Hu99,Thm. 3.11].Hence, H i pX t , Qq " / / H i pD t , Qq for i ă dimpD t q by the classical Lefschetz hyperplane theorem.Since the assertion is topological, this suffices to conclude. 14  The key step is the following result [Ab21, Prop.4.1].
Proposition 6.2 (Abugaliev).A smooth hypersurface D Ă X which is big and nef (or, equivalently, a smooth hypersurface with qpDq ą 0, cf.Proposition 3.3) cannot be covered by coisotropic varieties of codimension two in X.
Proof.Recall that a subvariety Z Ă X of codimension two is called coisotropic if the kernel of σ| Z : T Z / / Ω Z (over the smooth locus) is a sheaf of rank two, see Section 2.2.First observe that by Lemma 6.1, that for any subvariety Z Ă D there exists a class α P H 2 pX, Qq with α| D " rZs P H 2 pD, Qq.Clearly, the class α is of type p1, 1q.On the other hand, if Z Ă X is a coisotropic subvariety of codimension two, then 0 " rZs ^σn´1 P H 2n`2 pX, Cq.So, if Z Ă D is coisotropic and we write rZs " α| D P H 2 pD, Qq, then 0 " rDs ^α ^σn´1 P H 2n`2 pX, Cq, which implies ş X rDs ^α ^σn´1 ^σ n´1 " 0 and, therefore, qpD, αq " 0. Now use the well known formula (6.1) cf. [Hu02, Exer.23.2] to deduce from qpDq ą 0 and qpD, αq " 0, that 0 " ş X rDs 2n´1 ^α " ş Z rDs| 2n´2 Z .If D is ample, this is absurd.So we proved the stronger statement that an ample, smooth hypersurface D Ă X does not contain any coisotropic subvariety of codimension two. 15 If D is only big and nef, then qpD, αq " 0 still implies qpαq ă 0 by the Hodge index theorem, which in turn, by a formula [Ab21, Lem.4.2] similar to (6.1), gives ş X rDs 2n´2 ^α ^α ă 0. However, if D can be covered by coisotropic varieties of codimension two, then there exist two such Z 1 , Z 2 Ă D realising the same class α| D " rZ 1 s " rZ 2 s and then Y :" Z 1 XZ 2 Ă D is empty or of codimension two in D, which leads to the contradiction 0 ď Recall that a subvariety Z Ă D is called invariant under the characteristic foliation of the smooth hypersurface D if the leaf through any x P Z is contained in Z or, equivalently, if The following result [Ab21, Thm.0.5] is now a consequence of the above discussion.It concludes the proof of (iii) ñ (i) in Case (3).Theorem 6.3 (Abugaliev).Assume D Ă X is a smooth hypersurface of a hyperkähler manifold X satisfying qpDq ą 0. Then the generic leaf of the characteristic foliation on D is Zariski dense.
14 The original proof in [Ab21] uses the Kodaira-Akizuki-Nakano vanishing theorem.The above argument is quicker, but uses deformation theory and the projectivity criterion for hyperkähler manifolds. 15In dimension four this says that a smooth ample hypersurface does not contain any smooth Lagrangian surface, see Section 8.2.
Proof.If the generic leaf L Ă D is not Zariski dense, then its Zariski closure L Ă D defines a proper closed subvariety Z Ă D. The family of all such leaves gives a covering family tZ t u of D. Assume first that Z t Ă D is of codimension two in X.Since Z t " L is clearly invariant under the characteristic foliation and hence coisotropic by Corollary 2.6, (ii), this is a contradiction to Proposition 6.2.If the subvarieties Z t are of higher codimension, taking unions produces a covering family tZ 1 s u of D consisting of subvarieties of codimension two in X and such that each Z 1 s is a union of Z t " L. In particular, again by Corollary 2.6, (ii), each Z 1 s is coisotropic and one can conclude as before.6.5.(iv) ñ (i): Of course, this direction is a consequence of the implications proved before, but we wish to mention a weaker statement due to Hwang-Viehweg that motivated much of the later work on characteristic foliations.They proved [HV10, Thm.1.2] that the characteristic foliation of a smooth hypersurface D Ă X cannot be algebraic or, in other words, that dim L ą 1.

Alternative summary
We think it is instructive to present the discussion concerning the equivalence of the two conditions (iii) and (iv) in a somewhat differently structured way, making it more evident where and how foliations are used.‚ Assume qpDq ă 0. Then ş D c 1 pω D q ¨H2n´2 ă 0 and by [MM86, Miy87] D is uniruled.‚ Assume qpDq " 0. Then D and hence ω D are nef by Proposition 3.3.Using abundance conjecture for D combined with [DHP13], see footnote on page 12, one finds that D is semiample.Therefore, OpkDq defines a Lagrangian fibration f : X / / B for some k ą 0 and hence D " f ´1pf pDqq.‚ Assume qpDq ą 0. In this case D is of general type for which we presented two proofs: The one not using Proposition 3.3 just observed that under these assumptions D is in the interior of the pseudo-effective cone and hence big.‚ Assume D is uniruled.Then qpDq ă 0 is proved by excluding qpDq ą 0 and qpDq " 0. If qpDq ą 0, then D is of general type as explained above.To exclude qpDq " 0, one distinguishes two cases: First, if D is in the boundary of the movable cone, then ω D " OpDq| D is a limit of effective divisors and, therefore, pseudo-effective which contradicts the assumption that D foliation (of the smooth part) of D 0 : The leaves in the generic fibre π ´1pt0, tuq are the curves S 0 ˆx, x P S t , which in particular are not dense in the smooth(!) fibre S 0 ˆSt . 168.2.As we have just seen, if the hypersurface D Ă X is not smooth, then typically the conditions (i)-(iv) are not equivalent.
(i) Let us first discuss this in Case (2).Consider the pre-image f ´1p∆q Ă X of the discriminant divisor of a Lagrangian fibration f : X / / B. Note that even for ∆ irreducible its pre-image may be reducible.By [HO09, Thm.1.2] the characteristic foliation of any irreducible component of f ´1p∆q is algebraically integrable.Assume that there is a component of D of f ´1p∆q such that D " f ´1pf pDqq.This happens for instance when X is general among the hyperkähler manifolds with a Lagrangian fibration.Then, qpDq " 0 but its characteristic foliation is algebraically integrable.This divisor satisfies (iii) and (iv) of Case (2) but not (i).
(ii) We turn to Case (3).Consider a smooth cubic fourfold Y Ă P 5 and its Fano variety of lines X :" F pY q, which is a hyperkähler fourfold.The set of lines contained in a hyperplane section Y X H is a Lagrangian surface F pY X Hq Ă X which for generic H is smooth and of general type.For a one-dimensional family tY X H t u of hyperplane sections these Lagrangian surfaces sweep out a hypersurface D Ă X.Then qpDq ą 0, since for a generic cubic fourfold the Picard number of X " F pY q is one.
According to Corollary 2.6, (i), any leaf that intersects a generic F pY X H t q is contained in it.However, if D were smooth, then the results if Section 6.4 would imply that the generic leaf is dense.Contradiction.Hence, for no one-dimensional family of hyperplane section tY X H t u can the associated hypersurface D be smooth.
In particular this is an example of a singular divisor that satisfies (iii) and (iv) of Case (3) but not (i).
More abstractly, a smooth ample hypersurface D Ă X in a hyperkähler fourfold does not contain any Lagrangian surface, cf. the proof of Proposition 6.2.In particular, for a general cubic fourfold Y , a smooth Lagrangian surface F pY X H t q cannot be contained in any smooth divisor of X " F pY q.

‚
For any coherent subsheaf H Ă Ω D o {F one has κpD o {F, detpHqq ď 0. Indeed, according to [Miy87, Cor.8.6] or [MM86, Thm.1], the restriction of Ω D o {F to a generic complete intersection curve is semi-positive.At the same time, its determinant is trivial.Hence, all sub-sheaves of Ω D o {F have non-positive degree, which leads to the assertion.‚ Assuming g ą 1, there exists a coherent subsheaf H Ă Ω D o {F such that varpπ o q " κpD o {F, Hq, cf.[HV10, Thm.3.2 & Prop.4.4].Roughly, the relative cotangent sheaf of the natural map D o / / M g provides this sub-sheaf.(Strictly speaking, this is only true after passing to a finite cover of D o which does not affect the argument.) 7.1.(iii) ñ (iv): For a smooth hypersurface D Ă X one wants to show that the sign of qpDq largely determines the geometry of D.This part only involves more or less classical results and Proposition 3.3, i.e. the nefness of D if qpDq ě 0. Recall that the proof of Proposition 3.3 used foliations in an essential way.
7.2.(iv) ñ (iii): The geometry of D determines the sign of qpDq.Again, only Proposition 3.3 is used.