[Paper Review] Non-symplectic automorphisms of odd prime order on manifolds of $K3^{[n]}$-type

Chiara Camere, Alberto Cattaneo|arXiv (Cornell University)|Feb 1, 2018

Algebraic Geometry and Number Theory33 references3 citations

TL;DR

This paper classifies non-symplectic automorphisms of odd prime order on irreducible holomorphic symplectic manifolds of $K3^{[n]}$-type for all $n \geq 2$, using lattice-theoretic methods on the second cohomology lattice. It establishes that for $p^2 \nmid 2(n-1)$, the isometry class of the co-invariant lattice $S$ is uniquely determined by the triple $(p,m,a)$, and constructs explicit examples realizing all admissible actions for $n=3,4$, including a new construction on the Lehn–Lehn–Sorger–van Straten eightfold from cubic fourfold automorphisms.

ABSTRACT

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.

Motivation & Objective

• To extend the classification of non-symplectic automorphisms of odd prime order from $K3^{[2]}$-type to general $K3^{[n]}$-type manifolds for $n \geq 2$.
• To characterize the invariant and co-invariant lattices $T$ and $S$ inside the second cohomology lattice $H^2(X,\mathbb{Z}) \cong L$ via numerical invariants $p$, $m$, and $a$.
• To determine when the isometry class of $S$ is uniquely determined by the triple $(p,m,a)$, using Nikulin's lattice theory.
• To construct explicit automorphisms realizing all admissible cohomological actions, particularly for $n=3$ and $n=4$, including a new construction on the Lehn–Lehn–Sorger–van Straten eightfold.
• To correct and refine earlier classifications, especially those of non-symplectic involutions, by addressing errors in prior work and providing a systematic framework for odd prime orders.

Proposed method

• Leverages the global Torelli theorem for irreducible holomorphic symplectic manifolds, reducing the classification of automorphisms to the study of isometries on the second cohomology lattice $H^2(X,\mathbb{Z}) \cong L = U^{\oplus 3} \oplus E_8^{\oplus 2} \oplus \langle -2(n-1) \rangle$.
• Analyzes the action of a non-symplectic automorphism $\sigma$ of odd prime order $p$ via its induced isometry $\sigma^*$, focusing on the invariant lattice $T = H^2(X,\mathbb{Z})^{\sigma^*}$ and its orthogonal complement $S = T^\perp$.
• Uses the invariants $p$, $m$ (with $\mathrm{rk}(S) = (p-1)m$), and $a$ (with $H^2(X,\mathbb{Z})/T \oplus S \cong (\mathbb{Z}/p\mathbb{Z})^{\oplus a}$) to define admissible triples $(p,m,a)$ for each $n$
• Applies Nikulin's theory of lattices and the surjectivity of $O(S) \to O(q_S)$ to determine when the isometry class of $S$ is uniquely determined by $(p,m,a)$, particularly under the condition $p^2 \nmid 2(n-1)$.
• Constructs explicit automorphisms via geometric realizations: for $n=3,4$, examples are built using moduli spaces of twisted sheaves on $K3$ surfaces, and for $n=4$, a new construction arises from automorphisms of the underlying cubic fourfold on the Lehn–Lehn–Sorger–van Straten eightfold.
• Employs lattice-theoretic techniques to verify the existence and uniqueness of $S$ and $T$ for each admissible triple, and checks compatibility with the Beauville–Bogomolov–Fujiki form and the Néron–Tate pairing.

Experimental results

Research questions

• RQ1 Which triples $(p,m,a)$ of odd prime order $p$, rank $m$, and torsion parameter $a$ are admissible for non-symplectic automorphisms on $K3^{[n]}$-type manifolds for a given $n \geq 2$?
• RQ2 Under what conditions is the isometry class of the co-invariant lattice $S$ uniquely determined by the triple $(p,m,a)$?
• RQ3 Can all admissible cohomological actions of non-symplectic automorphisms of odd prime order be realized by explicit geometric constructions on $K3^{[n]}$-type manifolds?
• RQ4 How do automorphisms of the underlying cubic fourfold induce non-symplectic automorphisms on the Lehn–Lehn–Sorger–van Straten eightfold for $n=4$?
• RQ5 What corrections and refinements are needed for earlier classifications of non-symplectic involutions on $K3^{[n]}$-type manifolds?

Key findings

• For $p^2 \nmid 2(n-1)$, the isometry class of the co-invariant lattice $S$ is uniquely determined by the triple $(p,m,a)$, and under additional conditions on the discriminant group, the invariant lattice $T$ is also uniquely determined up to isometry of $L$.
• The classification of admissible triples $(p,m,a)$ is fully determined by lattice-theoretic constraints, with $p$ odd and $n \geq 2$, and the number of such triples increases significantly when $p$ divides $2(n-1)$.
• For $n=3$ and $n=4$, all admissible actions of non-symplectic automorphisms of odd prime order are realized by explicit geometric constructions, including on the Lehn–Lehn–Sorger–van Straten eightfold.
• A new construction of non-symplectic automorphisms of order 3 on the Lehn–Lehn–Sorger–van Straten eightfold is provided, arising from automorphisms of the underlying cubic fourfold.
• The paper corrects and refines earlier classifications of non-symplectic involutions, particularly those in [31], by identifying and rectifying errors and providing a consistent framework for odd prime orders.
• The table of lattices for $n=4$, $p=3$ lists 22 distinct configurations of $S$ and $T$, each corresponding to a unique admissible triple $(p,m,a)$, with labels indicating the structure of the lattices and their discriminant groups.