Characteristic foliation on hypersurfaces with positive Beauville-Bogomolov-Fujiki square
aa r X i v : . [ m a t h . AG ] F e b Characteristic foliation on hypersurfaces with positiveBeauville-Bogomolov-Fujiki square
Abugaliev RenatFebruary 5, 2021
Abstract
Let Y be a smooth hypersurface in a projective irreducible holomorphic symplecticmanifold X of dimension 2 n . The characteristic foliation F is the kernel of the symplec-tic form restricted to Y . In this article we prove that a general leaf of the characteristicfoliation is dense in Y if Y is nef and big. In this paper we are going to study the characteristic foliation F on a smooth hypersurface Y on a projective irreducible holomorphic symplectic manifold ( X, σ ) of dimension 2 n . Thecharacteristic foliation F on a hypersurface Y is the kernel of the symplectic form σ restrictedto T Y . If the leaves of a rank one foliation are quasi-projective curves, this foliation is calledalgebraically integrable. Jun-Muk Hwang and Eckart Viehweg showed in [14] that if Y is ofgeneral type, then F is not algebraically integrable. In [2] Ekaterina Amerik and Fr´ed´ericCampana completed this result to the following. Theorem 0.1 ([2]) . Let Y be a smooth hypersurface in an irreducible holomorphic symplecticmanifold X of dimension at least . Then the characteristic foliation on Y is algebraicallyintegrable if and only if Y is uniruled, i.e. covered by rational curves. The next step is to ask what could be the dimension of the Zariski closure of a generalleaf of F . In dimension 4 the situation is understood thanks to Theorem 0.2. Theorem 0.2 ([3]) . Let X be an irreducible holomorphic symplectic fourfold and let Y bean irreducible smooth hypersurface in X . Suppose that a general leaf of the characteristicfoliation F on Y is not algebraic, but there exists a meromorphic fibration p : Y C bysurfaces tangent to F . Then there exists an almost holomorphic Lagrangian fibration X B extending p . In particular, the Zariski closure of a general leaf is an abelian surface. This leads to the following conjecture.
Conjecture 0.3. (Campana) Let Y be a smooth hypersurface in an irreducible holomorphicsymplectic manifold X and let q be the Beauville-Bogomolov form on H ( X, Q ) . Then:1. If q ( Y, Y ) < , F is algebraically integrable and Y is uniruled. . If q ( Y, Y ) = 0 , the Zariski closure of a general leaf of F is an abelian surface ofdimension n ;3. If q ( Y, Y ) > , a general leaf of F is Zariski dense in Y ; The first case easily follows from [6, Theorem 4.2 and Proposition 4.7]. We refer to [2]for the details. The second case has been already studied by the author in [1]. Conjecturally, X is equipped with a rational Lagrangian fibration, and the hypersurface Y is the inverseimage of a hypersurface in its base. This conjecture was proved for manifolds of K3 type(see [4, Theorem 1.5]). Moreover a rational Lagrangian fibration can be replaced with aregular Lagrangian fibration, if we assume that the divisor Y is numerically effective. Thebase is conjectured to be P n , and this conjecture is known for n = 2 ([5] and [13]). In thatpaper we have considered an irreducible holomorphic symplectic manifold X equipped witha Lagrangian fibration π : X → P n and have obtained the result below. Theorem 0.4.
Let X be a projective irreducible holomorphic symplectic manifold and π : X → P n be a Lagrangian fibration. Consider a hypersurface D on P n such that itspreimage Y is a smooth irreducible hypersurface in X . Then the closure of a general leaf ofthe characteristic foliation on Y is a fiber of π (hence an abelian variety of dimension n ). In this work we study the third case of conjecture 0.3. To start we deal with an amplehypersurface (section 2). Next we want to extend this result. For this we need a versionof the Lefschetz hyperplane section theorem (proposition 3.1). In section 4 we obtain thefollowing result.
Theorem 0.5.
Let Y be a smooth nef and big hypersurface in an irreducible holomorphicsymplectic manifold X . Then a general leaf of the characteristic foliation of Y is Zariskidense in the hypersurface Y . In the end of this paper (section 5) we discuss the case of a not numerically effective hyper-surface. Unfortunately, we are not able to handle this case of the conjecture 0.3 completely.However, we settle it under some conditions (proposition 5.5).
Acknowledgments.
This research has been conducted under supervision of Ekaterina Amerik.The author is also very grateful to Emanuele Macri for a helpful discussion. The study hasbeen partially funded within the framework of the HSE University Basic Research Programand the Russian Academic Excellence Project ’5-100’.
In this section we recall some definitions and slightly reformulate conjecture 0.3. First, wedefine the characteristic foliation.
Definition 1.1.
Let Y be a hypersurface in X . Consider the restriction of T X to Y . Denoteits smooth locus by Y sm . The orthogonal complement of the bundle T Y sm in T X | Y sm isa line subbundle F of T Y sm ⊂ T X | Y sm . We call the rank one subbundle F ⊂ T Y sm the characteristic foliation . 2e recall that a regular foliation F is a subbundle of the tangent bundle T X satisfyingthe Frobenius condition [ F, F ] ⊂ F , which is trivial for line bundles. Locally a rank onefoliation is just a vector field. A subvariety is called invariant under the foliation F or F –invariant if it is tangent to this vector field. The Zariski closure of a leaf of F througha point x is the minimal F –invariant subvariety containing x . Finally, by the Zariski closureof a general leaf we mean the Zariski closure of a leaf of F through a very general point.One may reformulate the statement of third case of the conjecture 0.3 as follows (see [1] fora brief review on the foliations in the context of this problem). Conjecture 1.2.
Let Y be a smooth hypersurface of positive BBF square on an irreducibleholomorphic symplectic manifold X . Then there is no rational fibration Y B such thatits general fiber is tangent to F . In this section we prove conjecture 0.3 for a smooth and ample hypersurface Y . Theorem 2.1.
Let Y be a smooth and ample hypersurface in an irreducible holomorphicsymplectic manifold X . Then a general leaf of the characteristic foliation is Zariski dense in Y . In order to prove theorem 2.1 we assume the contrary. Let Y B be a fibrationsuch that its general fiber is invariant under the characteristic foliation F . Without loss ofgenerality one can assume that B is the projective line. Otherwise we replace Y B bythe composition Y B P , where B P is a pencil of hypersurfaces in B . Let Z be a fiber of this rational fibration. Definition 2.2.
A subvariety Z of codimension k is called coisotropic if the restriction ofthe symplectic form σ to the tangent space to Z at a general point has the least possiblerank n − k . Lemma 2.3.
Let ( X, σ ) be an irreducible holomorphic symplectic variety and Y be a smoothhypersurface in X . Consider a (possibly singular) subvariety Z of codimension 2 in X con-tained in Y (i.e. a hypersurface in Y ). The following statements are equals:1. The variety Z is invariant under the characteristic foliation;2. Z is coisotropic in ( X, σ ) (the restriction of σ to the smooth locus of Z has the leastpossible rank n − ).Proof. Let z be a smooth point of Z . Consider the vector spaces T Z,z ⊂ T Y,z ⊂ T X,z .= ⇒ Since Z is F Y -invariant, T Z,z contains T ⊥ Y,z . The line T ⊥ Y,z is orthogonal to any vector of T Z,z . Thus σ | T Z,z is degenerate. Since T Z,z has codimension 2, it is coisotropic. ⇐ = Since Z is coisotropic, T Z,z contains T ⊥ Z,z and hence it contains T ⊥ Y,z = F Y,z . As one may notice, for Z of higher codimension the first implication is wrong but the second implicationholds. Proposition 2.4.
Let Y be an ample smooth hypersurface in an irreducible holomorphicsymplectic manifold X . Then Y contains no coisotropic subvariety of codimension in X . In the rest of this section, we prove proposition 2.4. First we remark that being coisotropicis a cohomological property. In other words, a subvariety Z of codimension k (possiblysingular) is coisotropic if and only if [ Z ] ∪ σ n − k +1 = 0 ∈ H n +2 ( X, C ) (see [19, lemma 1.4]for the details). Now we use the ampleness of Y to apply the Lefschetz hyperplane theorem(LHT). It yields that there is a (not necessarily effective) divisor D , such that [ Z ] = [ D ] · [ Y ]. Lemma 2.5.
Let α, β ∈ NS( X ) . The class α · β is coisotropic if and only if q ( α, β ) = 0 .Proof. If Z is coisotropic [ Z ] ∪ σ n − = 0 and hence [ Z ] ∪ σ n − ¯ σ n − = α ∪ β ∪ σ n − ¯ σ n − = 0.By definition of BBF –form, if Z is coisotropic, q ( α, β ) = 0.So, D is orthogonal to Y . We show that it is impossible. Lemma 2.6.
Let α, β ∈ NS( X ) and q ( β, β ) > . Then the signs of q ( α, β ) q ( β, β ) n − and of αβ n − are the same.Proof. The Fujiki formula says that there is a positive constant c such that for every γ ∈ H ( X, Z ) q ( γ, γ ) n = cγ n . Let k be an integer. Applying the Fujiki formula for kβ + α we obtain the equality ofpolynomials in k : c ( kβ + α ) n = ( k q ( β ) + 2 q ( α, β ) k + q ( α )) n . (1)This equality of polynomials gives us the equality of the coefficients of the term of degree2 n −
1: 2 nc ( αβ n − ) = 2 nq ( α, β ) q ( β, β ) n − . Corollary 2.7.
Let Y be an ample divisor and let D be a divisor such that [ Z ] = [ Y ] · [ D ] isan effective Q –cycle. Then q ( Y, D ) > . In particular, Z is not coisotropic.Proof. Take α = [ D ] and β = [ Y ]. Since Z is effective and Y is ample we have[ Y ] n − ∪ [ D ] = [ Y ] n − ∪ [ Z ] > q ( D, Y ) >
0. 4
The LHT for a nef and big hypersurface in IHS
In the previous section we used the LHT for an ample hypersurface. In order to adapt thisproof for a nef and big hypersurface we show that the LHT holds for such.
Proposition 3.1.
Let Y be smooth, nef and big hypersurface in an irreducible holomorphicsymplectic manifold X . Then for i < dim X the restriction induces an isomorphism on thecohomology groups H i ( X, Q ) ∼ = H i ( Y, Q ) .Proof. First we are going to show that it is enough to prove the LHT for a smooth hypersur-face from the linear system | kY | for some integer k >>
0. Is is well-known (see [18, Chapter1.3]) that for a smooth hypersurface Y the LHT is equivalent to the Kodaira–Akizuki–Nakanovanishing for the line bundle O X ( Y ), which we are also going to prove. Proposition 3.2.
Let L be a nef and big line bundle on an irreducible holomorphic symplecticmanifold X . Then the Kodaira–Akizuki–Nakano vanishing (we will write it as the KAN forshortness) holds for L : H q ( X, Ω pX ⊗ L ∗ ) = 0 for p + q < dim X. Deligne and Illusie [9] prove that if the KAN holds for O X ( kY ) then it holds for O X ( Y ).Next we prove the LHT for a general hyperplane section from | kY | . By the Kawamata-Shokurov base-point-free theorem (see [7, Chapter 10]) the linear system | kY | induces aregular morphism φ to a projective space P N . By Bertini’s theorem ([15]) a general hy-persurface H from | kY | is smooth. Hence, the KAN is equivalent to the LHT for H . Wesummarize what we have written above in the following scheme. LHT f or H ∈ | kY | = ⇒ KAN f or O X ( kY ) = ⇒ KAN f or O X ( Y ) = ⇒ LHT f or Y. (2)This finishes the first step of the proof. In order to prove the LHT for H ∼ kY we recall thefollowing definition. Definition 3.3.
Let f : X → Y be a morphism of algebraic varieties. We call f semismall if for every subvariety Z of X , the following inequality holds2 dim Z ≤ dim X + dim f ( Z ) . We are going to prove that φ : X → P N is semismall in order to apply the followinggeneralization of the LHT ([10, Chapter 1.1]). Theorem 3.4.
Let f : X → P N be a (not necessarily proper) semismall map of a non-singular purely n -dimensional algebraic variety into complex projective space, and let H be ageneric hyperplane. Then H i ( X, f − ( H )) = 0 for i < n .Remark . Since | kY | is base-point-free, by the Bertini theorem a generic hyperplane sectionof | kY | is smooth. And all smooth hyperplane sections are diffeomorphic. We can replacethe word ”generic” with the word ”smooth” in theorem 3.4.5t remains to prove that φ : X | kY | −−→ P N is semismall. For this purpose we show that despitethe fact that Y is not ample, the hard Lefschetz theorem remains true for [ Y ] ∈ H ( X, Z )because Y deforms to an ample divisor. Lemma 3.6.
Let X be an irreducible holomorphic symplectic manifold of dimension n and Y be a hypersurface with q ( Y, Y ) > . Then for every < r < n we have an isomorphism ∪ [ Y ] n − r : H r ( X, Q ) ∼ −→ H n − r ( X, Q ) . Proof.
Let B M be a small analytic ball around the point corresponding to X in the modulispace of marked irreducible holomorphic symplectic manifolds (see [12]). Let B P be its imagein the period space. By [11, Theorem 8.1] the period map is surjective. The period space is anopen analytic subspace of the quadric q ( α, α ) = 0 in P (H ( X, C )). Consider the hyperplanesection H Y of this quadric defined by the equation q ( α, [ Y ]) = 0. The class [ Y ] is a Hodge classexactly in the Hodge structure corresponding to points of H Y . Moreover, for a general Hodgestructure A in H Y , the space of Hodge classes of A is generated by [ Y ] as a Q –vector space.Let M Y be the preimage of H Y in B M . Let X ′ be the holomorphic symplectic manifoldcorresponding to a general point of M Y . Since [ Y ] generates N S ( X ′ ) and its Beauville-Bogomolov square is positive, the class [ Y ] is ample on X ′ . Since H n ( X, Z ) = H n ( X ′ , Z ),lemma 3.6 follows from the hard Lefschetz theorem.Now we are ready to prove that φ : X → P N is semismall. Let Z be a closed subvarietyof X of codimension r . By lemma 3.6, [ Z ] ∪ [ Y ] n − r = 0 ∈ H n − r ( X, Z ). Hence the sectionof φ ( Z ) by a linear space of codimension 2 n − r in P N is not empty. Finally we obtain,dim φ ( Z ) ≥ n − r = 2(2 n − r ) − n = 2 dim Z − dim X. Remark . The connection between semismall maps and the Hard Lefschetz theorem wasalready been noticed in [8].
Remark . In [16, Remark 4.3.3] R. Lazarsfeld gives an example, where the KAN (or theLHT for a generic section) is not true for a nef and big line bundle. This line bundle is apull-back of O P (1) on the projective space P to the blowing-up Bl P P of this space at apoint P ∈ P . In this section we prove theorem 0.5. As it was shown in section 2 it is enough to prove thefollowing.
Proposition 4.1.
Let Y be a smooth nef and big hypersurface in an irreducible holomorphicsymplectic manifold X . Then Y can not be covered by a family of coisotropic subvarieties ofcodimension 2 in X . Y . Indeed when Y is not ample one may have Y n − · D = 0 (cf. proof of Corollary 2.7). But using that the family of coisotropic subvarietiescovers Y we can show that D · Y n − ≥
0. Indeed, let Z , Z be two different members ofthis family. Then Z ∩ Z is an effective cycle of codimension 4 in X . Thus the intersectionnumber Z · Z · Y n − = D · Y n − is not negative. This observation contradicts to thefollowing. Lemma 4.2.
In the assumption of lemma 2.6 if q ( β, β ) > , then α β n − < .Proof. The signature of the restriction of BBF form to H , ( X ) is (1 , h , ( X ) − α is negative. Considering the equality of polynomials (1) at the terms ofdegree 2 n − c n (2 n − α β n − = n ( n − q ( α, α ) q ( β, β ) n − . Since q ( α, α ) < q ( β, β ) >
0, the intersection number α β n − is negative. In this section we prove that a general leaf is also dense in Y if Y is not nef under someadditional condition on Y . A divisor with positive Beauville-Bogomolov-Fujiki square isnot necessarily nef. But it is birational to a nef divisor. The following proposition is aconsequence of theorem 1.2 and lemma 2.1 in [17]. Proposition 5.1.
Let X be an irreducible holomorphic symplectic manifold and Y be anirreducible hypersurface with q ( Y, Y ) > . Then there is an irreducible holomorphic symplecticmanifold X ′ with birational map ψ : X ′ X , such that the divisor Y ′ = ψ ∗ Y is nef. It is easy to see that if a general leaf of the characteristic foliation is dense in Y ′ , thenthe same is true for Y . Theorem 0.5 has the following corollary. Corollary 5.2.
In the assumptions of proposition 5.1, if Y ′ is smooth, then a general leaf ofthe characteristic foliation is dense in Y . Unfortunately, we do not know whether the smoothness of Y implies smoothness of Y ′ for an arbitrary Y . However, we are able to prove it under the assumption that Y ′ is veryample. Proposition 5.3.
Let ψ : X ′ X be a birational map of irreducible holomorphic symplecticmanifolds. Consider a smooth hypersurface Y in X . If its preimage Y ′ is very ample, then Y ′ is also smooth.Proof. Since ψ is an isomorphism in codimension 1, it induces an isomorphism of the linearsystems ψ ∗ : | Y | ∼ −→ | Y ′ | . Let ∆ be the set of singular divisors in | Y | and let ∆ ′ be the set ofsingular divisors in | Y ′ | . 7 emma 5.4. In the notation as above:1. ∆ ′ is an irreducible subvariety in | Y ′ | ;2. The image of ∆ ′ is contained in ∆ .Proof. Denote the linear space H ( X ′ , O X ′ ( Y ′ )) ∗ by V . The variety X ′ is embedded into P ( V ) and the discriminant ∆ ′ is embedded to P ( V ∗ ) = | Y ′ | . Consider the incidence variety I = { ( x, H ) ∈ X × P ( V ∗ ) | T X,x ⊂ T H,x } . This variety I is a projective bundle over X ′ .The fiber over a point x ∈ X is the linear system | Y ′ − x | ∼ = P dim V − dim X ′ consisting of thehyperplane sections singular at x . Hence, I is an irreducible variety. The variety ∆ ′ is theimage of I in P ( V ∗ ). Thus, ∆ ′ is also irreducible.The varieties X and X ′ are birational. Moreover, any divisor from the linear system | Y | isbirational to its preimage from | Y ′ | . A general singular divisor from ∆ ′ is singular outside ofthe indeterminacy locus of the birational map X ′ X . Thus, ∆ contains a general pointof φ ∗ (∆ ′ ) and by the closedness of ∆ contains φ ∗ (∆ ′ ).The second observation of lemma 5.4 implies that if Y ′ is singular, then Y is also singular.In other words, if Y is smooth, then Y ′ is also smooth.In conclusion, we have the following. Proposition 5.5.
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