Antisymplectic involutions of holomorphic symplectic manifolds

Beauville, Arnaud

doi:10.1112/jtopol/jtr002

Cited by 33 publications

(26 citation statements)

References 12 publications

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“…It follows from [, 3.4, Theorem 2; ] that there exists exactly one irreducible 20‐dimensional family of IHS fourfolds of

K 3^{[2]}

type which admit antisymplectic involutions. By , the invariant polarization in this family has Beauville degree

q = 2

and the quotient of such an involution for a generic element is a special sextic hypersurface in

P^{5}

called an EPW sextic.…”

Section: First Construction — Singular Epw Cubesmentioning

confidence: 99%

“…From there is only one possible invariant lattice of rank 2

H^{2} {(X, Z)}^{ι} : = false{x \in H^{2} (X, Z) false| ι^{*} (x) = x false}

that does not admit a polarization of Beauville degree

q = 2

, namely

U false(2 false) = [\begin{matrix} 0 & 2 \\ 2 & 0 \end{matrix}] .

Let

X

be an IHS fourfold with an involution and an invariant lattice

U (2)

. Then, by , the invariant lattice has signature

(1, 1)

for some

n

. In particular

X

is projective.…”

Section: First Construction — Singular Epw Cubesmentioning

confidence: 99%

“…Starting from this, we compute the invariants of

F (Z)

and

F_{0}

. The facts that

K_{F_{0}}^{2} = 288

and

χ ({scriptO}_{F_{0}}) = 37

follows from since

Y

moves in a 19‐dimensional family. By Noether's formula, we infer

c_{2} false(F_{0} false) = 156

.…”

Section: The Fano Surface Of the Verra Threefold Zmentioning

confidence: 99%

“…Recall that Beauville studied the invariants of the fixed loci of antisymplectic involutions on IHS fourfolds in general. In the case of 19-dimensional families of involutions on IHS fourfolds with b 2 = 23 it follows from [5,Theorem 2] that the invariants of the fixed locus F are K 2 F = 288 and χ(O F ) = 37. Using Proposition 4.6 we are able to deduce the invariants of a Hilbert schemes of conics on a Verra threefold Z.…”

Section: Propertiesmentioning

confidence: 99%

“…Let X be an IHS fourfold with an involution and an invariant lattice U (2). Then, by [5], the invariant lattice has signature (1, 1) for some n. In particular X is projective. Let h 1 and h 2 be the generators of the lattice with q(h 1 ) = q(h 2 ) = 0.…”

Section: First Construction -Singular Epw Cubesmentioning

confidence: 99%

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Hyper-Kähler fourfolds and Kummer surfaces

Iliev

Kapustka

et al. 2017

Proc. London Math. Soc.

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Abstract. We show that a Hilbert scheme of conics on a Fano fourfold double cover of P 2 × P 2 ramified along a divisor of bidegree (2, 2) admits a P 1 -fibration with base being a hyper-Kähler fourfold. We investigate the geometry of such fourfolds relating them with degenerated EPW cubes, with elements in the Brauer groups of K3 surfaces of degree 2, and with Verra threefolds studied in [Ver04]. These hyper-Kähler fourfolds admit natural involutions and complete the classification of geometric realizations of anti-symplectic involutions on hyper-Kähler 4-folds of type K3 [2] . As a consequence we present also three constructions of quartic Kummer surfaces in P 3 : as Lagrangian and symmetric degeneracy loci and as the base of a fibration of conics in certain threefold quadric bundles over P 1 .

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“…It follows from [, 3.4, Theorem 2; ] that there exists exactly one irreducible 20‐dimensional family of IHS fourfolds of

K 3^{[2]}

type which admit antisymplectic involutions. By , the invariant polarization in this family has Beauville degree

q = 2

and the quotient of such an involution for a generic element is a special sextic hypersurface in

P^{5}

called an EPW sextic.…”

Section: First Construction — Singular Epw Cubesmentioning

confidence: 99%

“…From there is only one possible invariant lattice of rank 2

H^{2} {(X, Z)}^{ι} : = false{x \in H^{2} (X, Z) false| ι^{*} (x) = x false}

that does not admit a polarization of Beauville degree

q = 2

, namely

U false(2 false) = [\begin{matrix} 0 & 2 \\ 2 & 0 \end{matrix}] .

Let

X

be an IHS fourfold with an involution and an invariant lattice

U (2)

. Then, by , the invariant lattice has signature

(1, 1)

for some

n

. In particular

X

is projective.…”

Section: First Construction — Singular Epw Cubesmentioning

confidence: 99%

“…Starting from this, we compute the invariants of

F (Z)

and

F_{0}

. The facts that

K_{F_{0}}^{2} = 288

and

χ ({scriptO}_{F_{0}}) = 37

follows from since

Y

moves in a 19‐dimensional family. By Noether's formula, we infer

c_{2} false(F_{0} false) = 156

.…”

Section: The Fano Surface Of the Verra Threefold Zmentioning

confidence: 99%

Section: Propertiesmentioning

confidence: 99%

Section: First Construction -Singular Epw Cubesmentioning

confidence: 99%

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Hyper-Kähler fourfolds and Kummer surfaces

Iliev

Kapustka

et al. 2017

Proc. London Math. Soc.

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Moduli Spaces of Cubic Threefolds and of Irreducible Holomorphic Symplectic Manifolds

Camere

2020

Birational Geometry and Moduli Spaces

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We extend the lattice-theoretic approach of Brandhorst-Cattaneo to classify algebraically trivial actions on the known IHS manifolds, up to deformation and birational conjugacy. In particular, we classify even order algebraically trivial non symplectic automorphisms, with or without trivial discriminant action. In the case of nontrivial discriminant actions, we show that such automorphisms exist only for finitely many known deformation types and orders.

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

Joumaah

2016

Math. Z.

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Abstract. 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.

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Antisymplectic involutions of holomorphic symplectic manifolds

Cited by 33 publications

References 12 publications

Hyper-Kähler fourfolds and Kummer surfaces

Hyper-Kähler fourfolds and Kummer surfaces

Moduli Spaces of Cubic Threefolds and of Irreducible Holomorphic Symplectic Manifolds

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

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