Irreducible symplectic varieties from moduli spaces of sheaves on K3 and Abelian surfaces

We show that the moduli spaces of sheaves on a projective K3 surface are irreducible symplectic varieties, and that the same holds for the fibers of the Albanese map of moduli spaces of sheaves on an Abelian surface

Surfaces with surjective endomorphisms of any given degree

We present a complete classification of complex projective surfaces X with nontrivial self-maps (i.e. surjective morphisms f:X→X which are not isomorphisms) of any given degree. The starting point of our classification are results contained in Fujimoto and Nakayama that provide a list of surfaces that admit at least one nontrivial self-map. We then proceed by a case by case analysis that blends geometrical and arithmetical arguments in order to exclude that certain prime numbers appear as degrees of nontrivial self-maps of certain surfaces

The Hodge numbers of O'Grady 10 via Ng\^o strings

We determine the Hodge numbers of the hyper-K\"ahler manifold known as O'Grady 10 by studying some related modular Lagrangian fibrations by means of a refinement of the Ng\^o Support Theorem.Comment: Revised and final version to appear in Jour. Math. Pur. et App

The Hodge numbers of O'Grady 10 via Ngô strings

We determine the Hodge numbers of the hyper-Kähler manifold known as O'Grady 10 by studying some related modular Lagrangian fibrations by means of Nĝo strings, which we introduce via a refinement of the Ngô Support Theorem

Holomorphic symplectic geometry: a problem list

A list of open problems on holomorphic symplectic, contact and Poisson manifolds

On the Beauville form of the known irreducible symplectic varieties

We study the global geometry of O’Grady’s ten-dimensional irreducible symplectic variety. We determine its second Betti number, its Beauville form and its Fujiki constant

Deformation of the O'Grady moduli spaces

In this paper we study moduli spaces of sheaves on an abelian or projective K3 surface. If S is a K3, v = 2w is a Mukai vector on S, where w is primitive and w(2) = 2, and H is a v-generic polarization on S, then the moduli space M-v of H-semistable sheaves on S whose Mukai vector is v admits a symplectic resolution (M) over tilde (v). A particular case is the 10-dimensional O'Grady example (M) over tilde (10) of an irreducible symplectic manifold. We show that (M) over tilde (v) is an irreducible symplectic manifold which is deformation equivalent to (M) over tilde (10) and that H-2 (M-v, Z) is Hodge isometric to the sublattice v(perpendicular to) of the Mukai lattice of S. Similar results are shown when S is an abelian surface

Factoriality properties of moduli spaces of sheaves on abelian and K3 surfaces

In this paper, we complete the determination of the index of factoriality of moduli spaces of semistable sheaves on an abelian or projective K3 surface S. If v is a Mukai vector and H a generic polarization, let Mv(S,H) be the moduli space of H-semistable sheaves on S with Mukai vector v. First, we describe in terms of v the pure weight-2 Hodge structure and the Beauville form on the second integral cohomology of the symplectic resolutions of M v(S,H) (when S is K3) and of the fiber Kv(S,H) of the Albanese map of Mv(S,H) (when S is abelian). Then, if S is K3, we show that Mv(S,H) is either locally factorial or 2-factorial, and we give an example of both cases. If S is abelian, we show that Mv(S,H) and Kv(S,H) are 2-factorial

On factoriality of threefolds with isolated singularities

We investigate the existence of complete intersection threefolds X ⊂ ℙn with only isolated, ordinary multiple points and we provide some sufficient conditions for their factoriality

Fine compactified Jacobians of reduced curves

To every singular reduced projective curve $X$ one can associate many fine compactified Jacobians, depending on the choice of a polarization on $X$, each of which yields a modular compactification of a disjoint union of a finite number of copies of the generalized Jacobian of $X$. We investigate the geometric properties of fine compactified Jacobians focusing on curves having locally planar singularities. We give examples of nodal curves admitting nonisomorphic (and even nonhomeomorphic over the field of complex numbers) fine compactified Jacobians. We study universal fine compactified Jacobians, which are relative fine compactified Jacobians over the semiuniversal deformation space of the curve $X$. Finally, we investigate the existence of twisted Abel maps with values in suitable fine compactified Jacobians