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Characteristic Foliation on Vertical Hypersurfaces in Holomorphic Characteristic foliation on vertical hypersurfaces in holomorphic symplectic manifolds with Lagrangian Fibration Abugaliev Renat December 9, 2020 Laboratoire de Math´ematiques d’Orsay, Univ. Paris-Sud, CNRS, Universit´eParis-Saclay, 91405 Orsay, France. E-mail address: [email protected] Abstract Let Y be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold X of dimension 2n. The characteristic foliation F is the kernel of the sym- plectic form restricted to Y . Assume that X is equipped with a Lagrangian fibration π : X → Pn and Y = π−1D, where D is a hypersurface in Pn. It is easy to see that the leaves of F are contained in the fibers of π. We prove that a general leaf is Zariski dense in a fiber of π. Introduction In this paper we are going to study the characteristic foliation F on a smooth hypersurface Y on a projective irreducible holomorphic symlpectic manifold (X, σ) of dimension 2n. The characteristic foliation F on a hypersurface Y is the kernel of the symplectic form σ restricted to TY . If the leaves of a rank one foliation are quasi-projective curves, this foliation is called arXiv:1909.07260v2 [math.AG] 8 Dec 2020 algebraically integrable. Jun-Muk Hwang and Eckart Viehweg showed in [14] that if Y is of general type, then F is not algebraically integrable. In paper [1] Ekaterina Amerik and Fr´ed´eric Campana completed this result to the following. Theorem 0.1. [1, Theorem 1.3] Let Y be a smooth hypersurface in an irreducible holomorphic symplectic manifold X of dimension at least 4. Then the characteristic foliation on Y is algebraically integrable 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 general leaf of F . In dimension 4 the situation is understood thanks to Theorem 0.2. 1 Theorem 0.2 ([2]). Let X be an irreducible holomorphic symplectic fourfold and let Y be an irreducible smooth hypersurface in X. Suppose that the characteristic foliation F on Y is not algebraically integrable, but there exists a meromorphic fibration on p : Y 99K C by surfaces invariant under F (see Definition 1.5). Then there exists a rational Lagrangian fibration X 99K 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. Let Y be a smooth hypersurface in an irreducible holomorphic symplectic manifold X and let q be the Beauville-Bogomolov form on H2(X, Q). Then: 1. If q(Y,Y ) > 0, a general leaf of F is Zariski dense in Y ; 2. If q(Y,Y )=0, the Zariski closure of a general leaf of F is an abelian variety of dimen- sion n; 3. If q(Y,Y ) < 0, F is algebraically integrable and Y is uniruled. The third case is easy. By [7, Theorem 4.2 and Proposition 4.7] Y is uniruled, if q(Y,Y ) < 0. There is a dominant rational map f : Y 99K W , such that the fibers of f are rationally connected (see for example [17, Chapter IV.5]). Rationally connected vari- eties do not have non-zero holomorphic differential forms. Thus, the form σ|Y is the pull-back 0 2 of some form ω ∈ H (W, ΩW ) and for any point x in Y the relative tangent space TY/W,x is the kernel of the form σ|Y . The kernel of σ|Y has dimension one at every point. So, the rational map f : Y 99K W is a fibration in rational curves and these rational curves are the leaves of the foliation F . For a hypersurface Y with non-negative Beauville-Bogomolov square the problem is more difficult. In this paper we are going to focus on the case where q(Y,Y ) = 0. Conjecturally, X admits a rational Lagrangian fibration, and hypersurface Y is the inverse image of a hypersur- face in its base. This conjecture was proved for manifolds of K3 type (see [3, Theorem 1.5]). Moreover a rational Lagrangian fibration can be replaced with a regular Lagrangian fibration, if we assume that the divisor Y is numerically effective. In this paper we consider an irre- ducible holomorphic symplectic manifold X equipped with a Lagrangian fibration π : X → B and assume that the base B is smooth. By [12], this means B ∼= Pn. Hwang and Oguiso in [13] prove that the characteristic foliation on the discriminant hypersurface is algebraically integrable, but this hypersurface is very singular. Our result is as follows. Theorem 0.4. Let X be a projective irreducible holomorphic symplectic manifold and π : X → Pn be a Lagrangian fibration. Consider a hypersurface D in Pn such that its preimage Y is a smooth irreducible hypersurface in X. Then the closure of a general leaf of the characteristic foliation on Y is a fiber of π (hence an abelian variety of dimension n). In order to prove Theorem 0.4 we are going to study the topology of Y . Theorem 0.5. Let X be a projective irreducible holomorphic symplectic manifold and π : X → Pn be a Lagrangian fibration. Consider a hypersurface D in Pn such that its preimage Y is a smooth irreducible hypersurface in X. Let Xb ⊂ Y be a smooth fiber of π, then the 2 2 morphism H (Y, Q) → H (Xb, Q) has rank one. 2 The paper is organized as follows. In subsection 1.1 we recall some results about La- grangian fibrations. In subsection 1.2 we recall some preliminaries on foliations and illustrate Theorem 0.4 on the examples in subsection 1.3. Next, in section 2 we study the topology of Y and prove Theorem 0.5. In section 3 we prove Theorem 0.4. In the end of the paper (Section 4) we discuss the question of smoothness of vertical hypersurfaces. Acknowledgments. This research has been conducted under supervision of Ekaterina Amerik. The author is also very grateful to Alexey Gorinov and Misha Verbitsky for the helpful dis- cussions and to Jorge Vit´orio Pereira for providing with the useful references on foliations. 1 Preliminaries 1.1 Lagrangian fibrations on irreducible holomorphic symplectic manifolds Let π : X → B be a Lagrangian fibration. The base B is conjectured to be always smooth and by [12] isomorphic to projective space. For n = 2 this conjecture was proved recently in [5] and [11]. In the present paper we assume this conjecture to be true, because the result below is crucial for our proof of theorem 0.5. Keiji Oguiso in [20] observed the following consequence of the results of [22] and [18]. Theorem 1.1 ([20],[22]). Let π : X → Pn be a Lagrangian fibration of a projective ir- reducible holomorphic symplectic manifold X and let Xb be a smooth fiber of π. Then 2 2 rank im(H (X, Q) → H (Xb, Q))=1. It is well-known that a smooth fiber of a Lagrangian fibration is an abelian variety. Concerning singular fiber there is the following result. Proposition 1.2. [13, Proposition 2.2] Let ∆ ⊂ Pn be the set of points b ∈ Pn such that the fiber Xb is singular. Then: 1. The set ∆ is a hypersurface in Pn. We call it the discriminant hypersurface. 2. The normalization of a general fiber of π over ∆ is smooth 3. The singular locus of a general singular fiber is a disjoint union of (n − 1)-dimensional complex tori. Remark 1.3. In particular, the set of points of X where the Lagrangian fibration π is singular is a subvariety of X of codimension 2. 1.2 Foliations and invariant subvarieties In this section we recall some definitions related to foliations and prove a preliminary result about the Zariski closure of the leaves. The results of this section are essentially contained in [8], [6] and [9], but they do not seem to be present in the literature in the form we need (corollary 1.11 and proposition 1.12). 3 Definition 1.4. Let X be a smooth algebraic variety. A foliation is an involutive (i.e. closed under the Lie bracket) subbundle F of the tangent bundle to X. A foliation is regular if the quotient sheaf TX /F is locally free. By Frobenius Theorem, locally X is a product Ls ×S (where Ls is an integral submanifold for s) and such decompositions into a product agree on intersection. Therefore the local integral curves glue together to a topological subspace of X. We call this subspace the leaf of F through the point x. In general, a leaf is not necessarily an algebraic subvariety. If all leaves of a foliation are algebraic we call this foliation algebraically integrable. For instance, let π : X → B be a smooth fibration. Obviously, TX/B is a regular foliation on X. Definition 1.5. One calls Y a smooth subvariety of X invariant under foliation F or F –invariant if TY contains F |Y . In other words, Y is invariant under F if it is a union of leaves. For the purpose of our work we need to extend these definitions, but we will consider only the rank one foliations. A rank one subsheaf of TX is automatically involutive. Definition 1.6. We denote by Sing(F ) the set of the singular points of F , i.e. the points such that the morphism F → TX has rank 0. It is easy to see that F is a regular foliation outside of Sing(F ).
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