# Hilbert schemes of K3 surfaces are dense in moduli

@article{Markman2011HilbertSO, title={Hilbert schemes of K3 surfaces are dense in moduli}, author={Eyal Markman and Sukhendu Mehrotra}, journal={arXiv: Algebraic Geometry}, year={2011} }

We prove that the locus of Hilbert schemes of n points on a projective K3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds is proven as well.

## 28 Citations

Mori cones of holomorphic symplectic varieties of K3 type

- Mathematics
- 2013

We determine the Mori cone of holomorphic symplectic varieties deformation equivalent to the punctual Hilbert scheme on a K3 surface. Our description is given in terms of Markman's extended Hodge…

Some remarks on moduli spaces of lattice polarized holomorphic symplectic manifolds

- Mathematics
- 2015

We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily--Borel compactification and investigate a…

Connected components of moduli spaces of generalised Kummer varieties

- Mathematics
- 2016

Moduli spaces of polarised (with fixed polarisation type) irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on $K3$ surfaces are not connected in…

On isotropic divisors on irreducible symplectic manifolds

- Mathematics
- 2013

Let X be an irreducible symplectic manifold and L a divisor on X. Assume that L is isotropic with respect to the Beauville-Bogomolov quadratic form. We define the rational Lagrangian locus and the…

On the monodromy of irreducible symplectic manifolds

- Mathematics
- 2014

Exploiting recent results on the ample cone of irreducible symplectic manifolds, we provide a different point of view for the computation of their monodromy groups. In particular, we give the final…

Bridgeland's stability and the positive cone of the moduli spaces of stable objects on an abelian surface

- Mathematics
- 2012

We shall study the chamber structure of positive cone of the albanese fiber of the moduli spaces of stable objects on an abelian surfaces via the chamber structure of stability conditions.

NO COHOMOLOGICALLY TRIVIAL NONTRIVIAL AUTOMORPHISM OF GENERALIZED KUMMER MANIFOLDS

- MathematicsNagoya Mathematical Journal
- 2018

For a hyper-Kähler manifold deformation equivalent to a generalized Kummer manifold, we prove that the action of the automorphism group on the total Betti cohomology group is faithful. This is a sort…

Integral cohomology of the generalized Kummer fourfold

- MathematicsAlgebraic Geometry
- 2018

We describe the integral cohomology of the Generalized Kummer fourfold giving an explicit basis, using Hilbert scheme cohomology and tools developed by Hassett and Tschinkel. Then we apply our…

Hodge theory and Lagrangian planes on generalized Kummer fourfolds

- Mathematics
- 2010

We analyze the intersection properties of projective planes embedded in generalized Kummer fourfolds, with a view toward classifying the homology classes represented by these submanifolds.

Polarized parallel transport and uniruled divisors on deformations of generalized Kummer varieties

- Mathematics
- 2016

In this note we characterize polarized parallel transport operators on irreducible holomorphic symplectic varieties which are deformations of generalized Kummer varieties. We then apply such…

## References

SHOWING 1-10 OF 23 REFERENCES

Any component of moduli of polarized hyperkaehler manifolds is dense in its deformation space

- Mathematics
- 2010

Let M be a compact hyperkaehler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in $H^2(M)$ defines a divisor $D_v$ in W consisting of all algebraic…

A simple proof of the surjectivity of the period map of K3 surfaces

- Mathematics
- 1981

The purpose of this note is to present a, simple proof of Todorov's theorem on the surjectivity of the period map of K3 surfaces. Our proof makes no use of any algebraic geometry results on the…

A survey of Torelli and monodromy results for holomorphic-symplectic varieties

- Mathematics
- 2011

We survey recent results about the Torelli question for holomorphicsymplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup WExc, of the…

Compact hyperkähler manifolds: basic results

- Mathematics
- 1997

The paper generalizes some of the well-known results for K3 surfaces to higher-dimensional irreducible symplectic (or, equivalently, compact irreducible hyperkaehler) manifolds. In particular, we…

The weight-two hodge structure of moduli spaces of sheaves on A K3 surface

- Mathematics
- 1995

We prove that the weight-two Hodge structure of mod- uli spaces of torsion-free sheaves on a K3 surface is as described by Mukai (the rank is arbitrary but we assume the first Chern class is…

Integral Symmetric Bilinear Forms and Some of Their Applications

- Mathematics
- 1980

We set up the technique of discriminant-forms, which allows us to transfer many results for unimodular symmetric bilinear forms to the nonunimodular case and is convenient in calculations. Further,…

The Kähler cone of a compact hyperkähler manifold

- Mathematics
- 2003

Abstract. This paper is a sequel to [11]. We study a number of questions only touched upon in [11] in more detail. In particular: What is the relation between two birational compact hyperkähler…

Prime exceptional divisors on holomorphic symplectic varieties and monodromy reflections

- Mathematics
- 2013

Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville-Bogomolov pairing. A divisor E on X is called a prime…

K3 surfaces via almost-primes

- Mathematics
- 2001

Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following fact: For any given positive…

On the monodromy of moduli spaces of sheaves on K3 surfaces II

- Mathematics
- 2003

Let S be a K3 surface. In part I of this paper, we constructed a representation of the group Aut D(S), of auto-equivalences of the derived category of S. We interpreted this infinite dimensional…