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

Kamenova, Ljudmila; Mongardi, Giovanni; Oblomkov, Alexei

doi:10.48550/arxiv.1809.02810

Cited by 2 publications

(2 citation statements)

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“…We notice that for M 3 , M 1 11 and M 2 11 we have that F ixpGq is given by isolated points (see Example 2 and Example 3 above), so they are not irreducible symplectic orbifolds. The fixed locus F ixpGq in the case of H n has codimension at least 4 by Theorem 1.1 of [24], so that H n is not an irreducible symplectic orbifold.…”

Section: Fujiki's Examplesmentioning

confidence: 99%

Examples of irreducible symplectic varieties

Perego

2019

Preprint

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Irreducible symplectic manifolds are one of the three building blocks of compact Kähler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are several singular analogues of irreducible symplectic manifolds, in particular in the context of compact Kähler orbifolds, and in the context of normal projective varieties with canonical singularities. In this paper we will collect their definitions, analyze their mutual relations and provide a list of known examples.

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Section: Fujiki's Examplesmentioning

confidence: 99%

Examples of irreducible symplectic varieties

Perego

2019

Preprint

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“…Similar techniques were adapted to the case of hyperkähler manifolds of K3 [n] type, and a classification of symplectic prime order automorphisms was achieved [Mon16]. Fixed loci of these involutions have also been identified [KMO18]. One can further adapt these techniques to the case of automorphisms of a cubic fourfold X.…”

Section: Introductionmentioning

confidence: 99%

Cubic Fourfolds with an Involution

Marquand¹

2022

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There are three types of involutions on a cubic fourfold; two of anti-symplectic type, and one symplectic. Here we show that cubics with involutions exhibit the full range of behaviour in relation to rationality conjectures. Namely, we show a general cubic fourfold with symplectic involution has no associated K3 surface and is conjecturely irrational. In contrast, a cubic fourfold with a particular anti-symplectic involution has an associated K3, and is in fact rational. We show such a cubic is contained in the intersection of all non-empty Hassett divisors; we call such a cubic Hassett maximal. We study the algebraic and transcendental lattices for cubics with an involution both lattice theoretically and geometrically.

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

Abstract: In this paper we describe the fixed locus of a symplectic involution on a hyperkähler manifold of type K3 [n] or of Kummer n type. We prove that the fixed locus consists of finitely many copies of Hilbert schemes of K3 surfaces of lower dimensions and isolated fixed points.

Cited by 2 publications

References 15 publications

Examples of irreducible symplectic varieties

Examples of irreducible symplectic varieties

Cubic Fourfolds with an Involution

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