Some remarks on moduli spaces of lattice polarized holomorphic symplectic manifolds

Camere, Chiara

doi:10.1142/s0219199717500444

Cited by 5 publications

(2 citation statements)

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“…In the present section we will describe the boundary components of F (N ) * for N ≤ 20. The Baily-Borel boundary components for N = 19 (quartic K3s) and N = 20 (EPW sextics modulo duality) have been described by Scattone [Sca87] and Camere [Cam15] respectively. We extend their analysis to cover the lower dimensional cases (this is necessary for our inductive study).…”

Section: Case Bmentioning

confidence: 99%

GIT Versus Baily-Borel Compactification for Quartic K3 Surfaces

Laza

O’Grady

2018

Geometry of Moduli

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In previous work, we have introduced a program aimed at studying the birational geometry of locally symmetric varieties of Type IV associated to moduli of certain projective varieties of K3 type. In particular, a concrete goal of our program is to understand the relationship between GIT and Baily-Borel compactifications for quartic K3 surfaces, K3's which are double covers of a smooth quadric surface, and double EPW sextics. In our first paper [LO16], based on arithmetic considerations, we have given conjectural decompositions into simple birational transformations of the period maps from the GIT moduli spaces mentioned above to the corresponding Baily-Borel compactifications. In our second paper [LO17] we studied the case of quartic K3's; we have given geometric meaning to this decomposition and we have partially verified our conjectures. Here, we give a full proof of the conjectures in [LO16] for the moduli space of K3's which are double covers of a smooth quadric surface. The main new tool here is VGIT for (2, 4) complete intersection curves. 7

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Section: Case Bmentioning

confidence: 99%

GIT Versus Baily-Borel Compactification for Quartic K3 Surfaces

Laza

O’Grady

2018

Geometry of Moduli

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“…Let now T be an even non-degenerate lattice of rank r ≥ 1 and signature (1, r − 1). A T -polarized IHS manifold is a pair (X, ι), where X is a projective IHS manifold and ι is a primitive embedding of lattices ι : T ֒→ NS(X) (see also [14]). Observe that we are then assuming that T has a primitive embedding in L, and we identify T with its image as sublattice of L.…”

Section: Non-symplectic Automorphisms Of Ihs Manifolds and (ρ T )-Pol...mentioning

confidence: 99%

Complex ball quotients from manifolds of K3[]-type

Boissière

Camere

Sarti

2019

Journal of Pure and Applied Algebra

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Cubic threefolds and hyperkähler manifolds uniformized by the 10-dimensional complex ball

2018

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We first prove an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of hyperkähler fourfolds of K3 [2] -type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is generated by a class of square 6; both these spaces are uniformized by the same 10-dimensional arithmetic complex ball quotient. We then study the degeneration of the automorphism along the loci of nodal or chordal degenerations of the cubic threefold, showing the birationality of these loci with some moduli spaces of hyperkähler fourfolds of K3 [2] -type with nonsymplectic automorphism of order three belonging to different families. Finally, we construct a cyclic Pfaffian cubic fourfold to give an explicit construction of a non-natural automorphism of order three on the Hilbert square of a K3 surface.Finally, in Section 5, looking at Hassett divisors on the moduli space of cubic fourfolds, we produce a Pfaffian cyclic cubic fourfold, thus we show: Corollary 1.3. There exists a smooth complex K3 surface whose Hilbert square admits a non-natural non-symplectic automorphism of order three, with fixed locus isomorphic to the Fano surface of a cubic threefold and invariant lattice isometric to 6 .We further explicitly describe this automorphism in terms of the Pfaffian geometry of the K3 surface, in analogy with Beauville's construction of the non-natural non-symplectic involution on the Hilbert scheme of a general quartic surface [10].Acknowledgements. The second author was partially supported by FIRB 2012 "Moduli spaces and their applications" and by "Laboratoire Internationale LIA LYSM". The authors thank Klaus Hulek, Shigeyuki Kondō, Christian Lehn, Gregory Sankaran and Davide Veniani for their helpful comments and suggestions. Fano varieties of lines of cyclic cubic fourfoldsWe work over the field of complex numbers.

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Some remarks on moduli spaces of lattice polarized holomorphic symplectic manifolds

Cited by 5 publications

References 30 publications

GIT Versus Baily-Borel Compactification for Quartic K3 Surfaces

GIT Versus Baily-Borel Compactification for Quartic K3 Surfaces

Complex ball quotients from manifolds of K3[]-type

Cubic threefolds and hyperkähler manifolds uniformized by the 10-dimensional complex ball

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