Symplectic birational transformations of finite order on O'Grady's sixfolds

Abstract

In this paper we consider irreducible holomorphic symplectic sixfolds of the sporadic deformation type discovered by O'Grady, and their symplectic birational transformations of finite order. We study the induced isometries on the Beauville-Bogomolov-Fujiki lattice, classifying all possible invariant and coinvariant sublattices, and providing explicit examples. As a consequence, we show that the isometry induced by a symplectic automorphism of finite order is necessarily trivial. This paper is a first step towards the classification of all finite groups of symplectic birational transformations.

Symplectic rigidity of O'Grady's tenfolds

“…A similar theorem holds for symplectic automorphisms of the other sporadic deformation type OG6 found by O'Grady in dimension 6 (see [4,Theorem 1.1]). On the other hand, Theorem 1.1 does not hold for birational transformations.…”

“…The monodromy group of a manifold of type OG10 coincides with the subgroup O + (L) of isometries of positive spinor norm [12]. [4,Lemma 2.3]). Therefore, a proof analogous to [4, Theorems 2.16 and 2.17] holds for the following theorems.…”

O'Grady tenfolds as moduli spaces of sheaves

“…In this section we give an application of the criterion about induced automorphisms given in Theorem 6.8. The lattice-theoretic criterion that we prove allows to determine if a birational transformation of a manifold of OG10 type which is at least birational to M v (S, θ), is induced by an automorphism of the K3 surface S. More precisely, by [11] we know that there are no regular symplectic involutions on a manifold of OG10 type, while by [21] we have a lattice-theoretic classification of birational symplectic involutions. We consider the classification of symplectic birational involutions on manifolds of OG10 type given in [21,Theorem 1.1], where the authors take a manifold of OG10 type, a fixed marking η : H 2 (X, Z) → L of X, and classify invariant and coinvariant sublattices, denoted by H 2 (X, Z) + ∼ = L G and H 2 (X, Z) − ∼ = L G respectively.…”

“…In the assumption of Proposition 6.4, we ask that the polarization θ ∈ NS(S) is ϕ-invariant, so that the automorphism of the K3 surface induces an automorphism of M v (S, θ). Note that this condition is never verified for a nontrivial symplectic automorphism of S according to the result of [11,Theorem 1.1]. On the other hand the case of induced nonsymplectic involutions will be the subject of an upcoming paper [5].…”

Birational geometry of irreducible holomorphic symplectic tenfolds of O'Grady type

“…The second ingredient in the proof of [Mon16] was an induced automorphism on the resolution M S (H) of M S (H), where S has a non trivial finite order symplectic automorphism ϕ and H is ϕ invariant. This is also an issue, because there is NO v-generic ϕ invariant polarization and NO finite order symplectic automorphism on a K3 inducing a regular automorphism on a manifold of OG10-type, as is also happening for OG6-type manifolds (see [GOV20]): Remark 6.2. Let S be a K3 surface and let ϕ ∈ Aut(S) be a finite order non trivial symplectic automorphism.…”