2014/05/02 by Amerik, Ekaterina, Campana, Frédéric
#14D99 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1405.0539
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 étale 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 Kähler case and apply this to the Lagrangian conjecture in the last section.