Moduli spaces of irreducible symplectic manifolds

Gritsenko, V. V.; Hulek, Klaus; Sankaran, G. K.

doi:10.1112/s0010437x0900445x

Cited by 101 publications

(167 citation statements)

References 75 publications

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“…X ∈ {F 1 , F 2 , N 3 }. Looking at the dimensions given by Table (8) we see that it remains to rule out one of the following inclusions:…”

Section: No Inclusion Relationsmentioning

confidence: 99%

“…Next, X N3 ⊂ B F2 cannot hold because the last row of Table ( 8) gives that 6 Boundary components meeting I in a subset of…”

Section: No Inclusion Relationsmentioning

confidence: 99%

“…Hyperkähler varieties belonging to a single deformation class break up into a countable family of polarized-deformation classes, indexed by the discrete invariants of the polarization such as the degree of its highest self-intersection, see [8] for more details.…”

Section: Introductionmentioning

confidence: 99%

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Moduli of double EPW-sextics

O’Grady¹

2016

Memoirs of the AMS

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We will study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of 3 C 6 modulo the natural action of SL6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics and hence birational to the moduli space of HK 4-folds of Type K3 [2] polarized by a divisor of square 2 for the Beauville-Bogomolov quadratic form. We will determine the stable points. Our work bears a strong analogy with the work of Voisin, Laza and Looijenga on moduli and periods of cubic 4-folds. We will prove a result which is analogous to a theorem of Laza asserting that cubic 4-folds with simple singularities are stable. We will also describe the irreducible components of the GIT boundary of M. Our final goal (not achieved in this work) is to understand completely the period map from M to the Baily-Borel compactification of the relevant period domain modulo an arithmetic group. We will analyze the locus in the GIT-boundary of M where the period map is not regular. Our results suggest that M is isomorphic to Looijenga's compactification associated to 3 specific hyperplanes in the period domain.

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“…X ∈ {F 1 , F 2 , N 3 }. Looking at the dimensions given by Table (8) we see that it remains to rule out one of the following inclusions:…”

Section: No Inclusion Relationsmentioning

confidence: 99%

“…Next, X N3 ⊂ B F2 cannot hold because the last row of Table ( 8) gives that 6 Boundary components meeting I in a subset of…”

Section: No Inclusion Relationsmentioning

confidence: 99%

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Moduli of double EPW-sextics

O’Grady¹

2016

Memoirs of the AMS

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“…[n] -type the modular group of the corresponding modular varieties is identical to a stable orthogonal group (see [GHS2]). The degree 2 extension of the stable orthogonal group changes the geometry of the modular varieties considerably.…”

Section: Any Totally Isotropic Subgroup Ofmentioning

confidence: 99%

“…[n] -type in [GHS2], where we restricted ourselves to the case of split polarisations (see [GHS1,Example 3.8] for a definition and details).…”

Section: Proposition 12 Every Component Mmentioning

confidence: 99%

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

Gritsenko

Hulek

Sankaran

2011

J. Differential Geom.

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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. IntroductionIrreducible symplectic manifolds are simply connected compact Kähler manifolds which have a (up to scalar) unique 2-form, which is non-degenerate. In dimension two these are the K3 surfaces. In higher dimension there are, so far, four known classes of examples. These are deformations of degree n Hilbert schemes of K3 surfaces (the K3[n] case), deformations of generalised Kummer varieties, and two examples of dimensions 6 and 10 due to O'GradyFrom the point of view of the Beauville lattice these examples fall into two series. The first consists of K3 surfaces, the K3[n] case and O'Grady's example of dimension 10. The Beauville lattices are the unimodular K3-lattice L K3 = 3U ⊕ 2E 8 (−1), the lattice L K3 ⊕ −2(n − 1) and L K3 ⊕ A 2 (−1). The moduli spaces of polarised irreducible symplectic manifolds of these classes are of dimensions 19, 20 and 21. The second series consists of generalised Kummer varieties and O'Grady's 6-dimensional variety with Beauville lattices 3U ⊕ −2 and 3U ⊕ −2 ⊕ −2 respectively. Here the dimensions of the moduli spaces of polarised varieties are 4 and 5.In order to describe moduli spaces of irreducible symplectic manifolds one must first classify the possible types of the polarisation. We do this in Section 3 for O'Grady's 10-dimensional example. As in the K3[n] case we find that we have a split and a non-split type. In this paper we shall mostly concentrate on the split case, when the modular group is maximal possible, but we shall also comment on the low degree non-split cases. 1In the non-split case we expect Kodaira dimension −∞ for the three cases of lowest Beauville degree, namely 2d = 12, 30, 48. For the next case of Beauville degree 2d = 66 we prove general type: see Corollary 4.3. The arguments used also suggest that 2d = 12, 30, 48 might be the only degrees of non-split polarisations giving unirational moduli spaces.We should like to comment that there is a natural series consisting of moduli of K3 surfaces of degree 2 (double planes branched along a sextic curve), the non-split K3[2] case of Beauville degree 2d = 6 (corresponding to cubic fourfolds and treated by Voisin in [Vo]) and O'Grady's example of dimension 10 with a non-split polarisation of degree 12. The lattices which are orthogonal to the polarisation vector in this series are 2U ⊕2E 8 (−1)⊕A n (−1) for n = 1, 2, 3. It would be very interesting to find a projective geometric realisation of O'Grady's 10-dimensional irreducible symplectic manifolds with non-split Beauville degree 12.In the split case we prove that the modular variety is...

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Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces

Harvey

Hassett

Tschinkel

2011

Comm Pure Appl Math

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We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic-symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformationequivalent to the Hilbert scheme of length-3 subschemes of a K3 surface. The class of the projective space in the cohomology ring has prescribed intersection properties, which translate into Diophantine equations. Possible homology classes correspond to integral points on an explicit elliptic curve; our proof entails showing that the only such point is two-torsion.

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Moduli spaces of irreducible symplectic manifolds

Cited by 101 publications

References 75 publications

Moduli of double EPW-sextics

Moduli of double EPW-sextics

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

Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces

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