In the present paper we prove that, on a hyperkähler manifold, walls of the Kähler cone and extremal rays of the Mori cone are determined by all divisors satisfying certain numerical conditions.

We realize O'Grady's six dimensional example of irreducible holomorphic symplectic manifold as a quotient of an IHS manifold of K3 [3] -type by a birational involution, thereby computing its Hodge numbers. IntroductionIn this paper we present a new way of obtaining O'Grady's six dimensional example of irreducible holomorphic symplectic manifold and use this to compute its Hodge numbers. Further applications, such as the description of the movable cone or the answer to Torelli-type questions for this deformation class of irreducible holomorphic symplectic manifolds, will be the topic of a subsequent paper. Recall that an irreducible holomorphic symplectic manifold (IHS) is a simply connected compact Kähler manifold that has a unique up to scalar holomorphic symplectic form. They arise naturally as one of the three building blocks of manifolds with trivial first Chern class according to the Beauville-Bogomolov decomposition [6], [2], the other two blocks being Abelian varieties and Calabi-Yau manifolds. By definition, IHS manifolds are higher dimensional generalizations of K3 surfaces, moreover they have a canonically defined quadratic form on their integral second cohomology group, which allows to speak of their periods and to develop their theory in a way which is analogous to the theory of K3 surfaces. The interested reader can see [16] and [39] for a general introduction on the topic. There are two deformation classes of IHS manifolds in every even dimension greater or equal to 4, introduced by Beauville in [2]. They are the Hilbert scheme of n points on a K3 and the generalized Kummer variety of dimension 2n of an abelian surface (i.e. the Albanese fiber of the Hilbert scheme of n+1 points of the abelian surface). Elements of these two deformation classes have second Betti number equal to 23 and 7, respectively, and are referred to as IHS manifolds of K3 [n] -type and of generalized Kummer type, respectively. There are two more examples, found by O'Grady in [37] and [38], of dimension ten and six, respectively, which are obtained from a symplectic resolution of some singular moduli spaces of sheaves on a K3 surface and on an abelian surface, respectively. They are referred to as the exceptional examples of IHS, and their deformation classes are denoted by OG10, respectively OG6. These exceptional examples have not been studied as much and their geometries are less understood. Though their topological Euler characteristic is known, see [42] and [31], even other basic invariants such as their Hodge numbers have not been computed yet. In the case of manifolds of K3 [n] -type, the Hodge numbers were computed by Göttsche [11]. One of the main results of this paper is to realize O'Grady's six dimensional example as a quotient of an IHS manifold of K3 [3] -type by a birational symplectic involution: we therefore relate this deformation class to the most studied deformation class of IHS manifolds and this allows us, by resolving the indeterminacy locus of the involution and by describing explicitly its fixed locus (wh...

We introduce the notion of induced automorphisms in order to state a criterion to determine whether a given automorphism on a manifold of K3 [n] -type is, in fact, induced by an automorphism of a K3 surface and the manifold is a moduli space of stable objects on the K3. This criterion is applied to the classification of non-symplectic prime order automorphisms on manifolds of K3 [2] -type and we prove that almost all cases are covered. Variations of this notion and the above criterion are introduced and discussed for the other known deformation types of irreducible symplectic manifolds. Furthermore we provide a description of the Picard lattice of several irreducible symplectic manifolds having a lagrangian fibration.

Let X be a Hyperkähler manifold deformation equivalent to the Hilbert square of a K3 surface and let ϕ be an involution preserving the symplectic form. We prove that the fixed locus of ϕ consists of 28 isolated points and 1 K3 surface, moreover the anti-invariant lattice of the induced involution on H 2 (X, Z) is isomorphic to E8(−2). Finally we prove that any couple consisting of one such manifold and a symplectic involution on it can be deformed into a couple consisting of the Hilbert square of a K3 surface and the involution induced by a symplectic involution on the K3 surface.

We construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure, and we link some of them to the projective families of irreducible holomorphic symplectic manifolds.