On the Characteristic Foliation on a Smooth Hypersurface in a Holomorphic Symplectic Fourfold

This terminology is explained by Bogomolov decomposition theorem which states that, up to a finite étale covering, each holomorphic symplectic manifold is a product of several irreducible ones and a torus. Let X be a holomorphic symplectic manifold with a holomorphic symplectic form ω. Let D be a smooth divisor on X. At each point of D, the restriction of ω to D has one-dimensional kernel. This gives a non-singular foliation F on D, called the characteristic foliation. Hwang and Viehweg in [HV] have shown that if X is projective and D is of general type, then F cannot be algebraic (unless in the trivial case when X is a surface, D is a curve, so F has a single leaf equal to D; recall that a foliation in curves on a compact Kähler manifold is called algebraic when all its leaves are compact complex curves). This result has been extended by Amerik and Campana in [AC2] to the case when D is not necessarily of general type; in particular, when X is irreducible, they proved that the characteristic foliation F on a nonsingular irreducible divisor D is algebraic if and only if the leaves of F are rational curves or X is a surface. Suppose now that X is an irreducible holomorphic symplectic fourfold and D, F are as above. It is easy to give an example when the Zariski closure of a general leaf of F is a surface. Indeed such is the case when X has a Lagrangian fibration f : X → Z and D is the preimage of a general curve C ⊂ Z. The leaves of F are contained in the fibers of f ; note that the general fiber is a torus by the Arnold-Liouville theorem. The aim of this paper is to prove that all examples where the Zariski closure of a general leaf is two-dimensional are obtained in this way.