Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type

Understand

Let X be a compact Kahler holomorphic-symplectic manifold, which is deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface.

• Let L be a nef line-bundle on X, such that the 2n-th power of c_1(L) vanishes and c_1(L) is primitive.
• Assume that the two dimensional subspace H^{2,0}(X) + H^{0,2}(X), of the second cohomology of X with complex coefficients, intersects trivially the integral cohomology.
• We prove that the linear system of L is base point free and it induces a Lagrangian fibration on X.