We prove that the characteristic foliation $F$ on a non-singular divisor $D$ in an irreducible projective hyperkaehler manifold $X$ cannot be algebraic, unless the leaves of $F$ are rational curves or $X$ is a surface. More generally, we show that if $X$ is an arbitrary projective manifold carrying a holomorphic symplectic $2$-form, and $D$ and $F$ are as above, then $F$ can be algebraic with non-rational leaves only when, up to a finite ��tale cover, $X$ is the product of a symplectic projective manifold $Y$ with a symplectic surface and $D$ is the pull-back of a curve on this surface. When $D$ is of general type, the fact that $F$ cannot be algebraic unless $X$ is a surface was proved by Hwang and Viehweg. The main new ingredient for our results is the observation that the canonical bundle of the (apriori, orbifold; but the orbifold structure is actually trivial) base of the family of leaves must be torsion. This implies, in particular, the isotriviality of the family of leaves of $F$. We also make some remarks in the K��hler case and apply this to the Lagrangian conjecture in the last section.
17 pages, LaTex 2e v2: minor corrections, a remark about a certain generalization added. v3: some arguments are added in order to make the application in section 5 work in the Kaehler case. v4: a simplification; indeed it turns out that the orbifold structure on the base is trivial. A paper on the isotriviality of families over a special orbifold base shall follow shortly