ON FIBRE SPACE STRUCTURES OF A PROJECTIVE IRREDUCIBLE SYMPLECTIC MANIFOLD

In this note, we investigate fibre space structures of a projective irreducible symplectic manifold. We prove that a 2n-cdimensional projective irreducible symplectic manifold admits only an n-dimensional fibration over a Fano variety which has only Q-factorial log-terminal singularities and whose Picard number is one. Moreover we prove that a general fibre is an abelian variety up to finite unramified cover, especially, for 4-fold, a general fibre is an abelian surface and all fibres are equidimensional.