Related papers: Birational symplectic manifolds and their deformat…

This article studies moduli spaces of Bridgeland semistable objects in the Kuznetsov component of a cubic fourfold that don't admit a symplectic resolution, i.e., moduli spaces of objects with non-primitve Mukai vector v=mv_0 that is not of…

In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree $0$ and rank $n$ on a compact Riemann surface $X$ of genus $g$. In particular, we prove that such moduli spaces are…

We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties.…

This is an improved version of the eprint previously entitled "Unexpected isomorphisms between hyperk\"ahler fourfolds." We study smooth projective hyperk\"ahler fourfolds that are deformations of Hilbert squares of K3 surfaces and are…

A symplectic manifold $(M,\omega)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and…

We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form. Such manifolda are also known as complex manifolds of hyperkaehler type. We obtain that a complex manifold of hyperkaehler type is Mirror…

Let $M$ be a hyperkahler manifold, $\Gamma$ its mapping class group, and $Teich$ the Teichmuller space of complex structures of hyperkahler type. After we glue together birationally equivalent points, we obtain the so-called birational…

We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the…

For a compact spinc manifold $X$ with boundary $b_1(\partial X)=0$, we consider moduli spaces of solutions to the Seiberg-Witten equations in a generalized double Coulomb slice in $L^2_1$ (i.e., $W^{1,2}$) Sobolev regularity. We prove they…

In general, if M is a moduli space of stable sheaves on X, there is a unique alpha in the Brauer group of M such that a pi_M^* alpha^{-1}-twisted universal sheaf exists on X times M. In this paper we study the situation when X and M are K3…

For any finite abelian group G, we study the moduli space of abelian $G$-covers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that, in the totally ramified case, the moduli space has…

We give a new construction of noncommutative surfaces via elliptic difference operators, attaching a 1-parameter noncommutative deformation to any projective rational surface with smooth anticanonical curve. The construction agrees with one…

Let $X$ be a K3 surface, and $H$ its primitive polarization of the degree $H^2=8$. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(2,H,2)$ is again a K3 surface, $Y$. In math.AG/0206158 we gave necessary and…

We construct the moduli of twisted sheaves on a projective variety. Then we generalize known results on the moduli space of usual sheaves on a K3 surface to the twisted case. Thus we consider the non-emptyness, the deformation type and the…

We provide a criterion for when Hilbert schemes of points on K3 surfaces are birational. In particular, this allows us to generate a plethora of examples of non-birational Hilbert schemes which are derived equivalent.

The present work is concerned with the study of four-dimensional irreducible holomorphic symplectic manifolds with second Betti number $23$. We describe their birational geometry and their relations to EPW sextics.

We study semistable pairs on elliptic K3 surfaces with a section: we construct a family of moduli spaces of pairs, related by wall crossing phenomena, which can be studied to describe the birational correspondence between moduli spaces of…

In the first part, Hyperkaehler Embeddings and Holomorphic symplectic Geometry I, we prove the following. Let $N$ be a closed analytic subvariety of a generic deformation of a holomorphically symplectic compact manifold $M$. Then the…

In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that, a surface admitting a smooth fibration as above is elliptic and…

Let X be a K3 surface of degree 8 in P^5 with hyperplane section H. We associate to it another K3 surface M which is a double cover of P^2 ramified on a sextic curve C. In the generic case when X is smooth and a complete intersection of…