Induced automorphisms on irreducible symplectic manifolds:

Mongardi, Giovanni; Wandel, Malte

doi:10.1112/jlms/jdv012

Cited by 30 publications

(56 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Using the exponential sequence we can find a B-field lift of α, i.e. a class B ∈ H 2 (Σ, Q) such that kB is integral and the map α : Tr(Σ) → Z/kZ is just the intersection product with kB (see [28, §3] In the case α = 0, which was already studied in [4] and [40], it is possible to provide some additional details on the action of induced automorphisms. Let v ∈ H * (Σ, Z) be a primitive, positive Mukai vector; then M v (Σ, 0) is isomorphic to the moduli space M τ (v) of τ -stable objects of Mukai vector v, for τ ∈ Stab(Σ) a v-generic Bridgeland stability condition on the derived category D b (Σ) (see [15] for details).…”

Section: Induced Automorphismsmentioning

confidence: 99%

“…For n ≥ 3, it is necessary to expand our pool of tools. Induced automorphisms on moduli spaces of (possibly twisted) sheaves on K3 surfaces, studied in [40] and [20], directly generalize natural automorphisms and allow to realize many new pairs (T, S): we show, in Section 5, how to apply these constructions when n = 3, 4. Admissible pairs (T, S) where T has rank one require special attention.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Non-symplectic automorphisms of odd prime order on manifolds of $$K3^{\left[ n\right] }$$-type

Camere

Cattaneo

2019

manuscripta math.

View full text Add to dashboard Cite

We classify non-symplectic automorphisms of odd prime order on irreducible holomorphic symplectic manifolds which are deformations of Hilbert schemes of any number n of points on K3 surfaces, extending results already known for n = 2. In order to do so, we study the properties of the invariant lattice of the automorphism (and its orthogonal complement) inside the second cohomology lattice of the manifold. We also explain how to construct automorphisms with fixed action on cohomology: in the cases n = 3, 4 the examples provided allow to realize all admissible actions in our classification. For n = 4, we present a construction of non-symplectic automorphisms on the Lehn-Lehn-Sorger-van Straten eightfold, which come from automorphisms of the underlying cubic fourfold.We remark that this is the first known geometric construction of a non-induced, non-symplectic automorphism of odd order on a manifold of K3 [4] -type. Moreover, thanks to it we are able to complete the list of examples of automorphisms of odd prime order p < 23 which realize all admissible pairs (T, S) for n = 3, 4.Acknowledgements. The authors thank Samuel Boissière, Andrea Cattaneo, Alice Garbagnati, Robert Laterveer and Giovanni Mongardi for many helpful discussions, as well as Christian Lehn and Manfred Lehn for their useful remarks and explanations. The authors are also extremely grateful to Alessandra Sarti and Bert van Geemen, for reading the paper and for their precious suggestions. Preliminary notions2.1. Lattices. Definition 2.1. A lattice L is a free abelian group endowed with a symmetric, non-degenerate bilinear form (·, ·) : L × L → Z. The lattice is even if the associated quadratic form is even on all elements of L. If t is a positive integer, L(t) denotes the lattice having as bilinear form the one of L multiplied by t. Examples of lattices,

show abstract

Section: Induced Automorphismsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-symplectic automorphisms of odd prime order on manifolds of $$K3^{\left[ n\right] }$$-type

Camere

Cattaneo

2019

manuscripta math.

View full text Add to dashboard Cite

show abstract

“…Moreover, they found that there is a unique 19‐dimensional irreducible family that admits the invariant lattice from each of the cases 1, 2, 4, respectively, and two families in the case 3. The families in the cases 1, 3, 4 admit polarizations of Beauville degree

q = 2

, it is not hard to see [, Remark 5.7] that they can be described as families of resolutions of special singular double EPW sextics.…”

Section: First Construction — Singular Epw Cubesmentioning

confidence: 99%

“…The description as double covers of Lagrangian degeneracy loci can also be applied to study degenerations of the family

scriptU

and permit to complete the classification of geometric realizations of automorphisms of IHS of type

K 3^{[2]}

given in . Note that as a direct consequence from [, § 5.1] we obtain the following.…”

Section: Introductionmentioning

confidence: 99%

“…The description as double covers of Lagrangian degeneracy loci can also be applied to study degenerations of the family

scriptU

and permit to complete the classification of geometric realizations of automorphisms of IHS of type

K 3^{[2]}

given in . Note that as a direct consequence from [, § 5.1] we obtain the following. Corollary Any IHS fourfold

X

of type

K 3^{[2]}

that admits non‐symplectic automorphism of prime order

p \neq 3, 23

is either in the closure of the family of double EPW sextics or in the closure of the family

scriptU

, or

X

is isomorphic to a moduli space of stable objects on a

K 3

‐surface and the automorphism is induced from an automorphism of the

K 3

‐surface.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hyper-Kähler fourfolds and Kummer surfaces

Iliev

Kapustka

et al. 2017

Proc. London Math. Soc.

View full text Add to dashboard Cite

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 .

show abstract

Moduli Spaces of Cubic Threefolds and of Irreducible Holomorphic Symplectic Manifolds

Camere

2020

Birational Geometry and Moduli Spaces

View full text Add to dashboard Cite

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.

show abstract

Induced automorphisms on irreducible symplectic manifolds:

Cited by 30 publications

References 26 publications

Non-symplectic automorphisms of odd prime order on manifolds of $$K3^{\left[ n\right] }$$-type

Non-symplectic automorphisms of odd prime order on manifolds of $$K3^{\left[ n\right] }$$-type

Hyper-Kähler fourfolds and Kummer surfaces

Moduli Spaces of Cubic Threefolds and of Irreducible Holomorphic Symplectic Manifolds

Contact Info

Product

Resources

About