Related papers: Birational symplectic manifolds and their deformat…

Using wall-crossing for K3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic Enriques surfaces for different stability conditions. As an application, we prove in the case of a Mukai vector of odd…

The present paper provides several results on automorphisms of hyperk\"ahler (or irreducible holomorphic symplectic) manifolds. In particular it focuses on the symplectic case and contains a classification of prime order symplectic…

We show that for many moduli spaces M of torsion sheaves on K3 surfaces S, the functor D(S) -> D(M) induced by the universal sheaf is a P-functor, hence can be used to construct an autoequivalence of D(M), and that this autoequivalence can…

We classify the group of birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one. We study whether these automorphisms are symplectic or non-symplectic and if there exists a hyperk\"ahler birational…

We give a necessary and sufficient condition for an automorphism of the Hilbert scheme of points on a K3 surface (non necessarily algebraic) to be induced by an automorphism of the surface. We prove furthermore that the group of birational…

We study birational maps among 1) the moduli space of semistable torsion sheaves of Hilbert polynomial $4m+2$ on a smooth quadric surface, 2) the moduli space of semistable torsion sheaves of Hilbert polynomial $m^{2}+3m+2$ on…

We prove a global Torelli theorem for the moduli space of marked triples (X,m,A), consisting of an irreducible holomorphic symplectic manifold X, a marking m of its second integral cohomology, and a stable and rigid sheaf A of Azumaya…

We introduce three non-compact moduli stacks parametrizing noncommutative deformations of Hirzebruch surfaces; the first is the moduli stack of locally free sheaf bimodules of rank 2, which appears in the definition of noncommutative…

We construct a finite dimensional Kaehler manifold with a holomorphic, symplectic circle action whose symplectic reduced spaces may be identified with the tau-vortex moduli spaces (or tau-stable pairs). The Morse theory of the circle action…

A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and…

We prove that the moduli spaces of K3 surfaces with non-symplectic involutions are unirational. As a by-product we describe configuration spaces of 4<d<9 points in the projective plane as arithmetic quotients of type IV.

We study moduli spaces of objects in the derived category of noncommutative ruled surfaces over orbifold curves to find equivariant deformations of moduli spaces of framed sheaves on equivariant elliptic surfaces. These derived categories…

We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of K\"ahler surfaces. This leads to a new proof of the deformation-invariance of…

These are notes of my lectures given at the school on intersection theory and moduli at the ICTP, Trieste. We construct moduli spaces of K3 surfaces and higherdimensional hyperkaehler manifolds, including moduli spaces of (2,2)-conformal…

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant forms of the invariant and…

We first prove an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of hyperkaehler fourfolds of K3^{[2]}-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is…

We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the…

A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally…

For infinitely many $d$, Hassett showed that special cubic fourfolds of discriminant $d$ are related to polarized K3 surfaces of degree $d$ via their Hodge structures. For half of the $d$, each associated K3 surface $(S,L)$ canonically…

We introduce the notion of induced birational transformations of irreducible holomorphic symplectic sixfolds of the sporadic deformation type discovered by O'Grady. We give a criterion to determine when a manifold of $OG_6$ type is…