Irreducible Holomorphic Symplectic Manifolds Research Articles

Symplectic rigidity of O’Grady’s tenfolds

We prove that any symplectic automorphism of finite order of an irreducible holomorphic symplectic manifold of O’Grady’s 10 10 -dimensional deformation type is trivial.

Characteristic Foliation on Hypersurfaces With Positive Beauville–Bogomolov–Fujiki Square

Abstract Let $Y$ be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold X of dimension 2n. The characteristic foliation $F$ is the kernel of the symplectic form restricted to Y. In this article, we prove that a generic leaf of the characteristic foliation is dense in Y if Y has positive Beauville–Bogomolov–Fujiki square.

Prime Order Isometries of Unimodular Lattices and Automorphisms of IHS Manifolds

Abstract We characterize conjugacy classes of isometries of odd prime order in unimodular ${\mathbb {Z}}$-lattices. This is applied to give a complete classification of odd prime order non-symplectic automorphisms of the known deformation types of irreducible holomorphic symplectic manifolds up to deformation and birational conjugacy.

Boucksom–Zariski and Weyl Chambers on Irreducible Holomorphic Symplectic Manifolds

Abstract Inspired by the work of Bauer, Küronya, and Szemberg, we provide for the big cone of a projective irreducible holomorphic symplectic (IHS) manifold a decomposition into chambers (which we describe in detail), in each of which the support of the negative part of the divisorial Zariski decomposition is constant. We see how the obtained decomposition of the big cone allows to describe the volume function. Moreover, similarly to the case of surfaces, we see that the big cone of a projective IHS manifold admits a decomposition into simple Weyl chambers, which we compare to that induced by the divisorial Zariski decomposition. To conclude, we determine the structure of the pseudo-effective cone.

Nonsymplectic automorphisms of prime order on O’Grady’s sixfolds

We classify nonsymplectic automorphisms of prime order on irreducible holomorphic symplectic manifolds of O’Grady’s 6-dimensional deformation type. More precisely, we give a classification of the invariant and coinvariant sublattices of the second integral cohomology group.

On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces

In this paper we prove a conjecture of Markman about the shape of the monodromy group of irreducible holomorphic symplectic manifolds of OG10 type. As a corollary, we also compute the locally trivial monodromy group of the underlying singular symplectic variety.

The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians

We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was previously announced in arXiv:1201.0031. Mongardi showed that the subgroup constructed here is in fact the whole monodromy group. As an application we prove the Hodge conjecture for the generic abelian fourfold of Weil type with complex multiplication by an arbitrary imaginary quadratic number field K, but only for polarizations with discriminant 1. The latter result is inspired by a recent observation of O'Grady that the third intermediate Jacobians of smooth projective varieties of generalized Kummer deformation type form complete families of abelian fourfolds of Weil type. Finally, we prove the surjectivity of the Abel-Jacobi map from the Chow group of co-dimension two algebraic cycles homologous to zero on every projective irreducible holomorphic symplectic manifold Y of Kummer type onto the third intermediate Jacobian of Y, as predicted by the generalized Hodge Conjecture.

Unobstructedness of hyperk\xe4hler twistor spaces

A family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.

Geometry and Automorphisms of Non-Kähler Holomorphic Symplectic Manifolds

AbstractWe consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works by D. Guan and the 1st author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$, which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction and prove that the automorphism group of $Q$ satisfies the Jordan property.

On the Definition of Irreducible Holomorphic Symplectic Manifolds and Their Singular Analogs

Abstract In the definition of irreducible holomorphic symplectic manifolds the condition of being simply connected can be replaced by vanishing irregularity. We discuss holomorphic symplectic, finite quotients of complex tori with ${\operatorname{h}}^0(X,\,\Omega ^{[2]}_X)=1$ and their Lagrangian fibrations. Neither $X$ nor the base can be smooth unless $X$ is a $2$-torus.

Fano Varieties of K3-Type and IHS Manifolds

AbstractWe construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure, and we link some of them to the projective families of irreducible holomorphic symplectic manifolds.

Twistor lines in the period domain of complex tori

As in the case of irreducible holomorphic symplectic manifolds, the period domain Compl of compact complex tori of even dimension 2n contains twistor lines. These are special 2-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori can be joined by a generic chain of twistor lines. We also prove a criterion of twistor path connectivity of loci in Compl where a fixed second cohomology class stays of Hodge type (1,1). Furthermore, we show that twistor lines are holomorphic submanifolds of Compl, of degree 2n in the Plucker embedding of Compl.

NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$-dimensional families of manifolds of $K3^{[n]}$-type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

Non-symplectic automorphisms of odd prime order on manifolds of $$K3^{\left[ n\right] }$$-type

We contribute to the classification of non-symplectic automorphisms of odd prime order on irreducible holomorphic symplectic manifolds which are deformations of Hilbert schemes of any number n of points on K3 surfaces, extending results already known for $$n=2$$ . In particular, we study the properties of the invariant lattice of the automorphism (and its orthogonal complement) inside the second cohomology lattice of the manifold. We also explain how to construct automorphisms with fixed action on cohomology: in the cases $$n=3,4$$ the examples provided realize all admissible actions in our classification. For $$n=4$$ , we present a construction of non-symplectic automorphisms on the Lehn–Lehn–Sorger–van Straten eightfold, which come from automorphisms of the underlying cubic fourfold.

Brane involutions on irreducible holomorphic symplectic manifolds

In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists’ language, that is, a submanifold which is either a complex or Lagrangian submanifold with respect to each of the three Kähler structures of the associated hyper-Kähler structure. Starting from a brane involution on a K3 or Abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier–Mukai transform. Later, we recall the lattice-theoretical approach to mirror symmetry. We provide two ways of obtaining a brane involution on the mirror, and we study the behavior of the brane involutions under both mirror transformations, giving examples in the case of a K3 surface and K3[2]-type manifolds.

On the existence of universal families of marked irreducible holomorphic symplectic manifolds

We prove the existence of a global family with natural universal properties over every component of the moduli space of marked irreducible holomorphic symplectic manifolds. The analogous result follows for Teichmüller spaces.

Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds

Rational curves on Hilbert schemes of points on K 3 K3 surfaces and generalised Kummer manifolds are constructed by using Brill–Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up to isometry, as dual divisors to such rational curves. The locus covered by the rational curves is then described, thus exhibiting algebraically coisotropic subvarieties. This provides strong evidence for a conjecture by Voisin concerning the Chow ring of irreducible holomorphic symplectic manifolds. Some general results concerning the birational geometry of irreducible holomorphic symplectic manifolds are also proved, such as a non-projective contractibility criterion for wall divisors.

Complex ball quotients from manifolds of K3[n]-type

We describe periods of irreducible holomorphic symplectic manifolds of K3[n]-type with a non-symplectic automorphism of prime order p≥3. These turn out to lie on complex ball quotients and we are able to give a precise characterization of when the period map is bijective by introducing the notion of K(T)-generality.

Some remarks on moduli spaces of lattice polarized holomorphic symplectic manifolds

We construct quasi-projective moduli spaces of [Formula: see text]-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily–Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.

The Hodge diamond of O’Grady’s six-dimensional example

We realize O’Grady’s six-dimensional example of an irreducible holomorphic symplectic (IHS) manifold as a quotient of an IHS manifold of$\text{K3}^{[3]}$type by a birational involution, thereby computing its Hodge numbers.