# Hyperk\"ahler manifolds

@article{Debarre2018HyperkahlerM, title={Hyperk\"ahler manifolds}, author={Olivier Debarre}, journal={arXiv: Algebraic Geometry}, year={2018} }

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

- MathematicsKyoto Journal of Mathematics
- 2023

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

- MathematicsLecture Notes of the Unione Matematica Italiana
- 2019

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…

### KODAIRA DIMENSION OF UNIVERSAL HOLOMORPHIC SYMPLECTIC VARIETIES

- MathematicsJournal 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

- MathematicsJournal de l’École polytechnique — Mathématiques
- 2020

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

- MathematicsMathematische Zeitschrift
- 2022

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

- MathematicsGeometry & Topology
- 2022

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

- Mathematics
- 2019

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

- MathematicsMathematische Zeitschrift
- 2021

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…

### ON THE CHOW RING OF CERTAIN LEHN–LEHN–SORGER–VAN STRATEN EIGHTFOLDS

- MathematicsGlasgow Mathematical Journal
- 2021

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…

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