Addendum To: On fibre space structures of a projective irreducible symplectic manifold
نویسنده
چکیده
Remark. Beauville proves that a Lagrangian fibration is a complete integrable system in [2, Proposition 1]. Thus, a general fibre of a fibre space of a projective irreducible symplectic manifold is an abelian variety. Remark. Markshevich states in [4, Remark 3.2] that there exists an irreducible symplectic manifold which has a family of non lagrangian tori. But this family does not form fibration.
منابع مشابه
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 an 2n-dimensional 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 un...
متن کاملHigher Direct Images of Dualizing Sheaves of Lagrangian Fibrations
We prove that the higher direct images of the dualizing sheaf of a Lagrangian fibration between smooth projective manifolds are isomorphic to the cotangent bundles of base space. As a corollary, we obtain that every Hodge number of the base space of a fibre space of an irreducible symplectic manifold is the same to that of a projective space if the base space is smooth.
متن کاملOn Nef Reductions of Projective Irreducible Symplectic Manifolds
Let X be a projective irreducible symplectic manifold and L is a non trivial nef divisor on X . Assume that the nef dimension of L is strictly less than dimX . We prove that L is semiample.
متن کاملA Special Class of Hyper-kähler Manifolds
We consider hyper-Kähler manifolds of complex dimension 4 which are fibrations. It is known that the fibers are abelian varieties and the base is P. We assume that the general fiber is isomorphic to a product of two elliptic curves. We prove that such a hyper-Kähler manifold is deformation equivalent to a Hilbert scheme of two points on a K3 surface. 1. Preliminaries First we define our main ob...
متن کاملHigher Dimensional Enriques Varieties with Even Index
Let Y be an Enriques variety of complex dimension 2n − 2 with n ≥ 2. Assume that n = 2m for odd prime m. In this paper we show that Y is the quotient of a product of a Calabi-Yau manifold of dimension 2m and an irreducible holomorphic symplectic manifold of dimension 2m − 2 by an automorphism of order n acting freely. We also show that both Y and its universal cover are always projective.
متن کامل