• Corpus ID: 119124909

Hyperk\"ahler manifolds

  title={Hyperk\"ahler manifolds},
  author={Olivier Debarre},
  journal={arXiv: Algebraic Geometry},
  • O. Debarre
  • Published 4 October 2018
  • Mathematics
  • arXiv: Algebraic Geometry
The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry: K3 surfaces and their higher-dimensional analogs, hyperkahler manifolds. These manifolds are interesting from several points of view: dynamical (some have interesting automorphism groups), arithmetical (although we will not say anything on this aspect of the theory), and geometric. It is also one of those rare cases where the Torelli theorem allows for a powerful link between the geometry of… 
9 Citations

On the period of Lehn, Lehn, Sorger, and van Straten’s symplectic eightfold

For the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane, we show that a natural Abel-Jacobi map from H^4_prim(Y) to H^2_prim(Z) is a Hodge

Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces

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


  • Shouhei Ma
  • Mathematics
    Journal of the Institute of Mathematics of Jussieu
  • 2021
Abstract We prove that the Kodaira dimension of the n-fold universal family of lattice-polarised holomorphic symplectic varieties with dominant and generically finite period map stabilises to the

Hilbert squares of K3 surfaces and Debarre-Voisin varieties

The Debarre-Voisin hyperk\"ahler fourfolds are built from alternating $3$-forms on a $10$-dimensional complex vector space, which we call trivectors. They are analogous to the Beauville-Donagi

On birational transformations of Hilbert schemes of points on K3 surfaces

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

Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties

We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these

Kodaira dimension of moduli of special $K3^{[2]}$-fourfolds of degree 2

We study the Noether-Lefschetz locus of the moduli space $\mathcal{M}$ of $K3^{[2]}$-fourfolds with a polarization of degree $2$. Following Hassett's work on cubic fourfolds, Debarre, Iliev, and

On vector bundles over hyperkähler twistor spaces

We study the holomorphic vector bundles E over the twistor space $${{\,\mathrm{Tw}\,}}(M)$$ Tw ( M ) of a compact simply connected hyperkähler manifold M . We give a characterization of the


Abstract We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional



A new six-dimensional irreducible symplectic variety

There are three types of “building blocks” in the Bogomolov decomposition [B, Th.2] of compact Kahlerian manifolds with torsion c1, namely complex tori, CalabiYau varieties, and irreducible

Automorphisms of hyperk\"ahler manifolds

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

Degenerations of cubic fourfolds and holomorphic symplectic geometry

In this thesis we study deformations of varieties of lines on smooth cubic hypersurfaces of the 5-dimensional complex projective space. These cubic hypersurfaces are also called cubic fourfolds.

On the Betti Numbers of Irreducible Compact Hyperkähler Manifolds of Complex Dimension Four

The study of higher dimensional hyperkahler manifolds has attracted much attention: we have [Wk], [Bg1,2,3,4], [Fj1,2], [Bv1], [Vb1,2], [Sl1,2], [HS], [Huy], [Gu3,4,5] etc. It is evident that there

Lectures on K3 Surfaces

Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular and each chapter ends with questions and open problems.

Projectivity and birational geometry of Bridgeland moduli spaces

We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a

Compact hyperkähler manifolds: basic results

Compact hyperkähler manifolds, or irreducible symplectic manifolds as they will be frequently called in these notes, are higher-dimensional analogues of K3 surfaces. That they indeed share many of

Morrison-Kawamata cone conjecture for hyperkahler manifolds

Let $M$ be a simple holomorphically symplectic manifold, that is, a simply connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$. We prove that the group of holomorphic

Addendum to "K3-surfaces of genus 8 and varieties of sums of powers of cubic fourfolds"

In this note, which is an addendum to the e-print math.AG/9810121, we prove that the variety VSP(F,10) of presentations of a general cubic form F in 6 variables as a sum of 10 cubes is a smooth