# Moscow Mathematical Journal

Volume 18, Issue 2, April–June 2018 pp. 193–204.

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

**Authors**:
E. Amerik (1) and L. Guseva (2)

**Author institution:**(1) National Research University Higher School of Economics, Laboratory of Algebraic Geometry and Applications, Usacheva 6, 119048 Moscow, Russia and

Université Paris-Sud, Laboratoire de Mathématiques d'Orsay, Campus Scientifique d'Orsay, Bât. 307, 91405 Orsay, France

(2) National Research University Higher School of Economics, Laboratory of Algebraic Geometry and Applications, Usacheva 6, 119048 Moscow, Russia

**Summary: **

Let *X* be an irreducible holomorphic symplectic fourfold
and *D* a smooth hypersurface in *X*. It follows from a result by E. Amerik
and F. Campana that the characteristic foliation (that is the foliation
given by the kernel of the restriction of the symplectic form to *D*) is not
algebraic unless *D* is uniruled. Suppose now that the Zariski closure of
its general leaf is a surface. We prove that *X* has a lagrangian fibration
and *D* is the inverse image of a curve on its base.

2010 Math. Subj. Class. 14D06, 14D15, 37F75

**Keywords:**Holomorphic symplectic manifolds, foliations, elliptic surfaces.

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