Irreducible Symplectic Manifolds Research Articles

Erratum: On the monodromy of irreducible symplectic manifolds \n (Algebraic Geometry 3 (2016), no. 3, 385–391)

Erratum: On the monodromy of irreducible symplectic manifolds \n (Algebraic Geometry 3 (2016), no. 3, 385–391)

A global Torelli theorem for singular symplectic varieties

We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky's work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geometry of symplectic manifolds, including some results on the classification of birational contractions of $K3^{[n]}$-type varieties.

Fujiki relations and fibrations of irreducible symplectic varieties

This paper concerns different types of singular complex projective varieties generalizing irreducible symplectic manifolds. We deduce from known results that the generalized Beauville-Bogomolov form satisfies the Fujiki relations and has the same rank as in the smooth case. This enables us to study fibrations of these varieties; imposing the newer definition from [GKP16, Definition 8.16.2] we show that they behave much like irreducible symplectic manifolds.

Stability of coisotropic fibrations on holomorphic symplectic manifolds

We investigate the stability of fibers of coisotropic fibrations on irreducible symplectic manifolds and generalize Voisin’s result on Lagrangian subvarieties to this framework. We present applications to such manifolds which are deformation equivalent to Hilbert schemes of points on a K3 surface or to generalized Kummer manifolds

Tautological sheaves: Stability, moduli spaces and restrictions to generalised Kummer varieties

Results on stability of tautological sheaves on Hilbert schemes of points are extended to higher dimensions and to the restriction of tautological sheaves to generalised Kummer varieties. This provides a big class of new examples of stable sheaves on higher dimensional irreducible symplectic manifolds. Some aspects of deformations of tautological sheaves are studied.

On the monodromy of irreducible symplectic manifolds

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

Non-symplectic involutions of irreducible symplectic manifolds of $$K3^{[n]}$$ K 3 [ n ] -type

This paper is concerned with non-symplectic involutions of irreducible symplectic manifolds of $K3^{[n]}$-type. We will give a criterion for deformation equivalence and use this to give a lattice-theoretic description of all deformation types. While moduli spaces of $K3^{[n]}$-type manifolds with non-symplectic involutions are not necessarily Hausdorff, we will construct quasi-projective moduli spaces for a certain well-behaved class of such pairs.

Induced automorphisms on irreducible symplectic manifolds:

We introduce the notion of induced automorphisms in order to state a criterion to determine whether a given automorphism on a manifold of K 3 [ n ] -type is, in fact, induced by an automorphism of a K 3 surface, and the manifold is a moduli space of stable objects on the K 3 . This criterion is applied to the classification of non-symplectic prime order automorphisms on manifolds of K 3 [ 2 ] -type, and we prove that almost all cases are covered. Variations of this notion and the above criterion are introduced and discussed for the other known deformation types of irreducible symplectic manifolds. Furthermore, we provide a description of the picard lattice of several irreducible symplectic manifolds having a lagrangian fibration.

Moduli spaces of polarized irreducible symplectic manifolds are not necessarily connected

We show that the moduli space of polarized irreducible symplectic manifolds of $K3^{[n]}$-type, of fixed polarization type, is not always connected. This can be derived as a consequence of Eyal Markman's characterization of polarized parallel-transport operators of $K3^{[n]}$-type.

Deformation of the O'Grady moduli spaces

In this paper we study moduli spaces of sheaves on an abelian or projective K3 surface. If S is a K3, v = 2w is a Mukai vector on S, where w is primitive and w(2) = 2, and H is a v-generic polarization on S, then the moduli space M-v of H-semistable sheaves on S whose Mukai vector is v admits a symplectic resolution (M) over tilde (v). A particular case is the 10-dimensional O'Grady example (M) over tilde (10) of an irreducible symplectic manifold. We show that (M) over tilde (v) is an irreducible symplectic manifold which is deformation equivalent to (M) over tilde (10) and that H-2 (M-v, Z) is Hodge isometric to the sublattice v(perpendicular to) of the Mukai lattice of S. Similar results are shown when S is an abelian surface.

Moduli Spaces of Polarized Symplectic O'Grady Varieties and Borcherds Products

We study moduli spaces of O'Grady’s ten-dimensional irreducible symplectic manifolds. These moduli spaces are covers of modular varieties of dimension 21, namely quotients of hermitian symmetric domains by a suitable arithmetic group. The interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group. This makes it different from both the K3 and the $K3^{[n]}$ case, which are of dimension 19 and 20 respectively.

Moduli spaces of irreducible symplectic manifolds

AbstractWe study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3[2]manifolds with polarisation of degree 2dand split type is of general type ifd≥12.

The 2-factoriality of the O’Grady moduli spaces

The aim of this work is to show that the moduli space M10 introduced by O’Grady is a 2-factorial variety. Namely, M10 is the moduli space of semistable sheaves with Mukai vector v: = (2, 0, −2) in \({H^{ev}(X,\mathbb{Z})}\) on a projective K3 surface X. As a corollary to our construction, we show that the Donaldson morphism gives a Hodge isometry between \({v^{\perp}}\) (sublattice of the Mukai lattice of X) and its image in \({H^{2} (\widetilde{M}_{10}, \mathbb{Z})}\), lattice with respect to the Beauville form of the 10-dimensional irreducible symplectic manifold \({\widetilde{M}_{10}}\), obtained as symplectic resolution of M10. Similar results are shown for the moduli space M6 introduced by O’Grady to produce its 6-dimensional example of irreducible symplectic variety.

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold $M$ of dimension $2n$, a projective manifold $X$ and a surjective holomorphic map $f:M \to X$ with connected fibers of positive dimension, we prove that $X$ is biholomorphic to the projective space of dimension $n$. The proof is obtained by exploiting two geometric structures at general points of $X$: the affine structure arising from the action variables of the Lagrangian fibration $f$ and the structure defined by the variety of minimal rational tangents on the Fano manifold $X$.

On monodromies of a degeneration of irreducible symplectic Kähler manifolds

We study the monodromy operators on the Betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middle cohomology is at least the half of the dimension. This implies that the “mildest” singular fiber of a good degeneration with non-trivial monodromy of irreducible symplectic manifolds is quite different from the generic degeneration of abelian varieties or Calabi–Yau manifolds.

On nef reductions of projective irreducible symplectic manifolds

Let X be a projective irreducible symplectic manifold and L be a non trivial nef divisor on X. Assume that the nef dimension of L is strictly less than the dimension of X. We prove that L is semiample.

Higher direct images of dualizing sheaves of Lagrangian fibrations

Let f : X → S be a Lagrangian fibration between projective varieties. We prove that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] ≅ Ω i S if S is smooth. Suppose that X is an irreducible symplectic manifold or a certain moduli space of semistable torsion free sheaves on a K 3 surface, the Hodge numbers satisfy h p,q ( S ) = h p,q [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /], where n = dim S . If S ≅ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] and X is an irreducible symplectic manifold, there exists a hypersurface M f of the Kuranishi space of X such that every member of the Kuranishi family over M f admits a Lagrangian fibration over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /].

Holomorphic Symplectic Manifolds and Lagrangian Fibrations

We introduce the geometrical nature of fibre space structures of an irreducible symplectic manifold and holomorphic Lagrangian fibrations.

Counter-example to global Torelli problem for irreducible symplectic manifolds

A simply connected compact Kaehler manifoldX is an irreducible symplectic manifold if there is an everywhere non-degenerate holomorphic 2-form Ω on X with H0(X,Ω2X) = C[Ω]. By definition, X has even complex dimension. There is a canonical symmetric form qX on H (X,Z), which is called the BeauvilleBogomolov form (cf. [Be]). On the other hand, since X is Kaehler, H(X,Z) has a natural Hodge structure of weight 2. If two irreducible symplectic manifolds X and Y are bimeromorphically equivalent, then there is a natural Hodge isometry between (H(X,Z), qX) and (H (Y,Z), qY ) (cf. [O, Proposition (1.6.2)], [Huy, Lemma 2.6]). Debarre [De] has constructed bimeromorphically equivalent irreducible symplectic manifolds X and Y such that X and Y are not isomorphic. This is a counter-example to the following problem.

On fibre space structures of a projective irreducible symplectic manifold, Topology 38, (1999) 79–83

On fibre space structures of a projective irreducible symplectic manifold, Topology 38, (1999) 79–83