Related papers: Integral Transforms and Deformations of K3 Surface…

We study the deformations of a smooth curve $C$ on a smooth projective threefold $V$, assuming the presence of a smooth surface $S$ satisfying $C \subset S \subset V$. Generalizing a result of Mukai and Nasu, we give a new sufficient…

We study autoequivalences and stability conditions on the derived category of coherent sheaves on a singular surface $X$ which arises as an open subvariety of a type III Kulikov degeneration of K3 surfaces. The surface $X$ consists of four…

We examine the space of surfaces in $\RR^{3}$ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the space $\Mk$ of…

Let $X$ be the Fano threefold of index one, degree $22$, and $\mathrm{Pic}(X)\cong\mathbb{Z}$. Such a threefold $X$ can be realized by a regular zero section $\mathbf{s}$ of $(\bigwedge^2\mathcal{F}^{*})^{\oplus 3}$ over Grassmannian…

Let k be a field of arbitrary characteristic and let Q be a quiver of finite representation type. In this paper we prove that if M is an indecomposable kQ-module then the universal deformation ring of M over kQ is isomorphic to k.

Let M be a K3 surface or an even-dimensional compact torus. We show that the category of coherent sheaves on M is independent from the choice of the complex structure, if this complex structure is generic.

In this paper, we investigate Keller's deformed Calabi--Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an $n$-dimensional smooth variety $Y$, we describe the derived category of the total…

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.

Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on…

We characterize the subscheme of the moduli space of torsion-free sheaves on an elliptic surface which are stable of relative degree zeero (with respect to a polarization of type aH+bf, H being the section and f the elliptic fibre) which is…

We show that some Gieseker stable sheaves on a projective K3 surface $X$ are stable with respect to a stability condition of Bridgeland on the derived category of $X$ if the stability condition is in explicit subsets of the space of…

Let X and Y be compact hyper-Kahler manifolds deformation equivalence to the Hilbert scheme of length n subschemes of a K3 surface. A cohomology class in their product XxY is an analytic correspondence, if it belongs to the subring…

We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: 1. We describe the nef cone, the movable cone, and the effective…

We give a vanishing and classification result for holomorphic differential forms on smooth projective models of the moduli spaces of pointed K3 surfaces. We prove that there is no nonzero holomorphic k-form for 0<k<10 and for even k>19. In…

Let $X$ be a complex smooth quasi-projective surface acted upon by a finite group $G$ such that the quotient $X/G$ has singularities only of ADE type. We obtain an explicit expression for the generating series of the Euler characteristics…

For a real algebraic K3 surface $X(R)$, we give all possible values of the dimension $h^1_{alg}(X(R)$ of the group $\H^1_{alg}(X(R),Z/2)$ of algebraic cycles of $X(R)$. In particular, we prove that if $X(R)$ is not an M-surface, $X(R)$ can…

Let $X$ be a smooth connected projective algebraic curve over an algebraically closed field, and let $S$ be a finite nonempty closed subset in $X$. We study deformations of $\overline{\mathbb F}_\ell$-sheaves. The universal deformation…

The main result of the present paper is a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized projective surfaces. For this, we use the theory of good moduli spaces, whose study was initiated by Alper.…

Let $(X,L_{X})$ be an $n$-dimensional polarized manifold. Let $D$ be a smooth hypersurface defined by a holomorphic section of $L_{X}$. We prove that if $D$ has a constant positive scalar curvature K\"{a}hler metric, $X \setminus D$ admits…

We survey recent advances in the theory of moduli spaces of stable sheaves on hyperk\"ahler manifolds of dimension greater than $2$. We start by recalling the well-known theory in dimension $2$, i.e.~for $K3$ surfaces, emphasizing the…