A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known (due to Huybrechts) that a given compact manifold admits onlyâ€¦ (More)

We prove finiteness of the deformation classes of hyperkÃ¤hler Lagrangian fibrations in any fixed dimension with fixed Fujiki constant and discriminant of the Beauville-Bogomolov-Fujiki lattice. Weâ€¦ (More)

A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which followsâ€¦ (More)

In this paper we classify the possible degenerate fibers which can occur in a semistable degeneration of two-dimensional tori under the assumption that the canonical bundle of the total space of theâ€¦ (More)

Let p : M â†’ B be a Lagrangian fibration on a hyperkÃ¤hler manifold of maximal holonomy (also known as IHS), and H be the generator of the Picard group of B. We prove that pâˆ—(H) is a primitive class onâ€¦ (More)

We consider hyper-KÃ¤hler 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 toâ€¦ (More)

This paper is a survey of finiteness results in hyperkÃ¤hler geometry. We review some classical theorems by Sullivan, KollÃ¡r-Matsusaka, Huybrechts, as well as theorems in the recent literature byâ€¦ (More)

Every fibration of a projective hyper-KÃ¤hler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified byâ€¦ (More)