Local projectivity of Lagrangian fibrations on Hyperkähler manifolds

Abstract

We show that if $$f:X\rightarrow B$$ is a Lagrangian fibration from a compact connected hyperkahler (and thus Kahler) manifold X onto a projective normal variety B, then f is locally projective. This answers a question raised by L. Kamenova and strengthens a former result (see Campana in Math Z 252:147–156, 2005, Proposition 2.1), according to which the smooth fibres of f are projective.