On natural deformations of symplectic automorphisms of manifolds of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>K</mml:mi><mml:msup><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup></mml:math> type

Mongardi, Giovanni

doi:10.1016/j.crma.2013.07.020

Cited by 18 publications

(17 citation statements)

References 7 publications

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“…As explained after Remark 2 of [38], we can find (Y, ι ′ ) which is deformation equivalent to (X, ι) such that there exist two marks ϕ, ϕ ′ and a generic complex torus A with P(Y, ϕ) = P(K 2 (A), ϕ ′ ).…”

Section: Uniqueness and Fixed Locusmentioning

confidence: 98%

Integral cohomology of the Generalized Kummer fourfold

Kapfer¹,

Menet²

2016

Preprint

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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 results to a IHS variety with singularities, obtained by a partial resolution of the generalized Kummer fourfold quotiented by a symplectic involution. We calculate the Beauville-Bogomolov form of this new variety, presenting the first example of such a form that is odd.

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Section: Uniqueness and Fixed Locusmentioning

confidence: 98%

Integral cohomology of the Generalized Kummer fourfold

Kapfer¹,

Menet²

2016

Preprint

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“…A pair (Y, H) is called numerically standard if the representation of H on H 2 (Y, Z) coincides with that of a standard pair (X, H), up to the action of the monodromy group. More specifically, there exists a K3 (or abelian) surface S with an H action such that This definition is slightly stronger than the one given in [10] for manifolds of K3 [n] type , but they coincide when n − 1 is a prime power, which was the case of interest in that paper. Notice that it is relatively easy to check the first two conditions, while the third one is more involved but often unnecessary, see Proposition 4.13.…”

Section: Preliminariesmentioning

confidence: 99%

“…The moduli space of pairs consisting of a hyperkähler manifold of K3 [n] type together with a symplectic involution is connected. This follows from the following result of the second named author [10,Theorem 2.5]. Here we include a more general version of the result where we remove the assumption that n − 1 is a prime power.…”

Section: Fixed Loci Of Symplectic Involutionsmentioning

confidence: 99%

Symplectic involutions of $K3^{[n]}$ type and Kummer $n$ type manifolds

Kamenova,

Mongardi,

Oblomkov

2018

Preprint

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“…By [33,Theorem 1.6], there exists a unique element w ∈ W Exc (X) and a birational map β ∈ Bir(X) such that η −1 • ρ • η = w • β * . The map β is unique in this case since the natural map Aut(X) → O(H 2 (X, Z)) is injective in this deformation class (see [35,Lemma 1.2]).…”

Section: Degenerationsmentioning

confidence: 99%

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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On natural deformations of symplectic automorphisms of manifolds of K3[n] type

Abstract: In the present paper we prove that finite symplectic groups of automorphisms of manifolds of K3 [n] type can be obtained by deforming natural morphisms arising from K3 surfaces if and only if they satisfy a certain numerical condition.

Cited by 18 publications

References 7 publications

Integral cohomology of the Generalized Kummer fourfold

Integral cohomology of the Generalized Kummer fourfold

Symplectic involutions of $K3^{[n]}$ type and Kummer $n$ type manifolds

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

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