Related papers: Hyperk\"ahler manifolds

We construct a new 20-dimensional family of algebraic hyper-Kaehler fourfolds and prove that they are deformation-equivalent to the second punctual Hilbert scheme of a K3 surface of degree 22.

Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces…

Using the theory of holes of the Leech lattice and Borcherds method for the computation of the automorphism group of a K3 surface, we give an effective bound for the set of isomorphism classes of projective models of fixed degree for…

We introduce and study the notion of atomic sheaves and complexes on higher-dimensional hyper-K\"ahler manifolds and show that they share many of the intriguing properties of simple sheaves on K3 surfaces. For example, we prove formality of…

We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two…

We give a characterisation of Atiyah's and Hitchin's transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperk\"ahler manifolds arising from the…

We prove the Noether-Lefschetz conjecture on the moduli space of quasi-polarized K3 surfaces. This is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal…

Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a…

We give a constructive proof of the Hodge conjecture for complex $K3$ surfaces that does not rely on Torelli-type results. Starting with an arbitrary rational $(1,1)$-class $\alpha\in H^{1,1}(X,\mathbb{Q})$, we algorithmically build a…

In this paper almost complex surfaces of the nearly K\"ahler $S^3\times S^3$ are studied in a systematic way. We show that on such a surface it is possible to define a global holomorphic differential, which is induced by an almost product…

We remark the density of the jumping loci of the Picard number of a hyperk\"ahler manifold under small one-dimensional deformation and provide some applications for the Mordell-Weil groups of Jacobian K3 surfaces.

The aim of this paper is to derive explicitly a connection between the Zagier elliptic trilogarithm and Mahler measures of a certain family of three-variable polynomials defining K3 surfaces. In addition, we prove some linear relations…

We study the derived categories of twisted supersingular K3 surfaces. We prove a derived crystalline Torelli theorem for twisted supersingular K3 surfaces, characterizing Fourier-Mukai equivalences in terms of isomorphisms between their…

We survey the basic theory of non-commutative K3 surfaces, with a particular emphasis to the ones arising from cubic fourfolds. We focus on the problem of constructing Bridgeland stability conditions on these categories and we then…

M-theory suggests the study of 11-dimensional space-times compactified on some 7-manifolds. From its intimate relation to superstrings, one possible class of such 7-manifolds are those that have Calabi-Yau threefolds as boundary. In this…

We investigate the collapsing geometry of hyperk\"ahler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperk\"ahler metrics on the K3 manifold is isometric to one of…

In this paper, we extend our geometrical derivation of expansion coefficients of mirror maps by localization computation to the case of toric manifolds with two K\"ahler forms. Especially, we take Hirzebruch surfaces F_{0}, F_{3} and…

This article grew out of an effort to understand the smooth mapping class groups of certain 4-manifolds in a geometric manner. We prove a smooth analog of the Birman-Hilden theorem for manifolds that admit a hyperk\"ahler structure. This…

We generalize Nikulin's and Dolgachev's lattice-theoretical mirror symmetry for K3 surfaces to lattice polarized higher dimensional irreducible holomorphic symplectic manifolds. In the case of fourfolds of $K3^{\left[2\right]}-$type we then…

We review recent developments in the arithmetic of K3 surfaces. Our focus lies on aspects of modularity, Picard number and rational points. Throughout we emphasise connections to geometry.