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Let $M$ be a hyperkahler manifold of maximal holonomy (that is, an IHS manifold), and let $K$ be its Kahler cone, which is an open, convex subset in the space $H^{1,1}(M, R)$ of real (1,1)-forms. This space is equipped with a canonical bilinear symmetric form of signature $(1,n)$ obtained as a restriction of the Bogomolov-Beauville-Fujiki form. The set of vectors of positive square in the space of signature $(1,n)$ is a disconnected union of two convex cones. The positive cone is the component which contains the Kahler cone. We say that the Kahler cone is round if it is equal to the positive cone. The manifolds with round Kahler cones have unique bimeromorphic model and correspond to Hausdorff points in the corresponding Teichmuller space. We prove thay any maximal holonomy hyperkahler manifold with $b_2 > 4$ has a deformation with round Kahler cone and the Picard lattice of signature (1,1), admitting two non-collinear integer isotropic classes. This is used to show that all known examples of hyperkahler manifolds admit a deformation with two transversal Lagrangian fibrations, and the Kobayashi metric vanishes unless the Picard rank is maximal.
Let $(M,I, Omega)$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $pi:; M mapsto X$, and $eta$ a closed form of Hodge type (1,1)+(2,0) on $X$. We prove that $Omega:=Omega+pi^* eta$ is again a holomorphically
We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalenc
Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a sub
The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry: K3 surfaces and their higher-dimensional analogs, hyperkahler manifolds. These manifolds are interesting from several points of view: dynamical (so
In previous papers we define certain Lagrangian shadows of ample divisors in algebraic varieties. In the present brief note an existence condition is discussed for these Lagrangian shadows.