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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...

The aim of this paper is to study the singularities of certain moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non-generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. For sheaves that are pure of dimension one, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety and that, via this isomorphism, the natural symplectic resolutions correspond to variations of GIT quotients of the quiver variety.

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Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-Kähler setting, we can then deduce a finiteness statement for monodromy acting on H 2 , once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the geometry of hyper-Kähler manifolds, we prove that a finite monodromy projective degeneration of hyper-Kähler manifolds has a smooth filling (after base change and birational modifications). As a consequence of these two results, we prove a generalization of Huybrechts' theorem about birational versus deformation equivalence, allowing singular central fibers. As an application, we give simple proofs for the deformation type of certain geometric constructions of hyper-Kähler manifolds (e.g. Debarre-Voisin [DV10] or Laza-Saccà-Voisin [LSV17]). In a slightly different direction, we establish some basic properties (dimension and rational homology type) for the dual complex of a Kulikov type degeneration of hyper-Kähler manifolds.Date: October 25, 2018.