Characteristic foliation on non-uniruled smooth divisors on hyperkähler manifolds.

*(English)*Zbl 1402.14010Summary: We prove that the characteristic foliation \(\mathcal{F}\) on a non-singular divisor \(\mathcal{D}\) in an irreducible projective hyperkähler manifold \(X\) cannot be algebraic, unless the leaves of \(\mathcal{F}\) 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 \(\mathcal{D}\) and \(\mathcal{F}\) are as above, then \(\mathcal{F}\) can be algebraic with non-rational leaves only when, up to a finite étale cover, \(X\) is the product of a symplectic projective manifold \(Y\) with a symplectic surface and \(\mathcal{D}\) is the pullback of a curve on this surface. When \(\mathcal{D}\) is of general type, the fact that \(\mathcal{F}\) 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 class of the (orbifold) base of the family of leaves is zero. This implies, in particular, the isotriviality of the family of leaves of \(\mathcal{F}\). We show this, more generally, for regular algebraic foliations by curves defined by the vanishing of a holomorphic \((d-1)\)-form on a complex projective manifold of dimension \(d\).

##### MSC:

14D06 | Fibrations, degenerations in algebraic geometry |

14C05 | Parametrization (Chow and Hilbert schemes) |

14J28 | \(K3\) surfaces and Enriques surfaces |

14J70 | Hypersurfaces and algebraic geometry |

32S65 | Singularities of holomorphic vector fields and foliations |

53C26 | Hyper-Kähler and quaternionic Kähler geometry, “special” geometry |