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Corrigendum to "Nonsymplectic automorphisms of prime order on O'Grady's sixfolds"

This paper serves as a corrigendum to a previous study on nonsymplectic automorphisms of prime order on O'Grady's sixfolds by correcting the lattice-theoretic classification to include previously missing cases and explaining their recovery.

Original authors: Annalisa Grossi, Stevell Muller

Published 2026-07-28

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CC BY 4.0

Original authors: Annalisa Grossi, Stevell Muller

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe of shapes not as the smooth curves of a beach ball or the sharp angles of a pyramid, but as a vast, invisible landscape of "hyper-Kähler manifolds." Think of these as ultra-complex, multi-dimensional sculptures that exist in the realm of pure mathematics. They are special because they possess a hidden, perfect symmetry—a kind of internal rhythm that mathematicians call a "symplectic structure." Now, imagine trying to twist or rotate one of these invisible sculptures. Sometimes, you can spin it in a way that keeps its internal rhythm perfectly intact; other times, the twist breaks the rhythm, creating a "nonsymplectic" effect.

Mathematicians are obsessed with cataloging these twists. Why? Because just as a crystal's structure determines how it shines or breaks, the way these shapes can be twisted reveals deep secrets about the fundamental building blocks of geometry. In this specific corner of math, there is a famous, complex shape known as an "OG6 type" manifold. It's like a rare, six-dimensional diamond. For a long time, researchers thought they had a complete map of all the possible ways to twist this diamond without breaking its core structure. They believed they had found every possible "twist pattern" (called a conjugacy class) that could exist. But, as with any big map, it turns out a few territories were left blank, and a few landmarks were drawn in the wrong place.

This paper is a "corrigendum," which is a fancy word for a polite but necessary correction. The authors, Annalisa Grossi and Stevell Muller, are essentially saying, "We found some missing pages in our map and fixed a few typos." They went back to the OG6 diamond and used a powerful computer program called OSCAR to re-scan the landscape. Their main finding is that there are exactly 45 distinct ways to twist this shape with a prime number of steps (like 2, 3, 5, or 7 steps) while keeping it effective. The original map had missed several of these twists and included a few duplicates.

To understand the fix, imagine you are organizing a massive library of books. The original team had sorted the books by their cover designs (the "invariant sublattices") and the empty spaces left behind when you took a book out (the "coinvariant sublattices"). They thought they had every combination. However, Grossi and Muller discovered that they had missed a few rare book covers entirely. For instance, they found a missing combination where the "cover" was a specific lattice shape called $U \oplus 3 \oplus [-2]$ and the "empty space" was $U \oplus [2]$ . Crucially, the authors clarify that the lattice $U \oplus 3 \oplus [-2]$ itself was actually already present in the original tables; the error was that the specific pairing of this lattice with the coinvariant lattice $U \oplus [2]$ had been overlooked. This wasn't just a small oversight; it meant the total count of unique twist patterns was incomplete.

The paper also acts like a detective cleaning up a messy crime scene. They found that some entries in their original tables were actually the same case disguised in different clothing. Using a mathematical trick where they swapped parts of the lattice (like realizing that a $[2] \oplus [-4]$ block is mathematically identical to a $[-2] \oplus [4]$ block), they realized that several entries in their original list were duplicates. They removed these double-counts and replaced them with the correct, unique cases. They also found cases where a twist was possible but had been overlooked, marked in their new tables with a blue row or a special "trefoil" symbol (♣) to show that this specific twist was a new discovery.

Crucially, the authors didn't just guess these numbers; they proved them. They used a rigorous, step-by-step algorithm implemented in their computer program to systematically check every possibility. They didn't just suggest that there might be more twists; they demonstrated that there are exactly 45 conjugacy classes of these effective nonsymplectic groups. They also corrected a specific error in their previous work where they claimed a certain combination of shapes was possible, only to realize later that no such shape could actually exist in the OG6 universe.

In the end, this paper is a story of precision. It's about taking a complex, abstract map of a six-dimensional world and making sure every single dot is in the right place. By adding the missing cases and removing the duplicates, the authors have provided the math community with the definitive, corrected list of how these special shapes can be twisted. It's a reminder that even in the most abstract corners of mathematics, the details matter, and sometimes, the most important discovery is finding the pieces you forgot to pack in the first place.

Technical Summary: Corrigendum to "Nonsymplectic automorphisms of prime order on O'Grady's sixfolds"

Problem Statement
This note addresses a classification gap in the lattice-theoretic study of nonsymplectic automorphisms of prime order acting on hyper-Kähler manifolds of OG6 type (O'Grady's sixfolds). In the original work [3], the authors attempted to classify the invariant and coinvariant sublattices associated with effective nonsymplectic groups $G \subset O(L)$ , where $L \cong U^{\oplus 3} \oplus [-2]^{\oplus 2}$ is the second integral cohomology lattice equipped with the Beauville–Bogomolov–Fujiki quadratic form. The original classification was incomplete, containing missing cases for specific pairs of invariant and coinvariant sublattices $(L^G, L_G)$ and, in some instances, erroneous entries or duplicates in the associated tables.

Methodology
The authors employ a computational approach using the computer algebra system OSCAR [2], specifically utilizing algorithms developed by Brandhorst and Hofmann [1]. The classification strategy relies on the following steps:

Reduction to Isometries: Following [3, Proposition 3.4], the problem is reduced to classifying finite subgroups $G \subset O(L)$ of prime order $p \in \{2, 3, 5, 7\}$ such that the invariant sublattice $L^G$ has signature $(1, *)$ .
Generator Conditions: The authors identify representatives of conjugacy classes of isometries $g \in O(L)$ $g \in O (L)$ satisfying two conditions:
- The invariant lattice $L^g$ has signature $(1, *)$ .
- The real quadratic space $\ker((g + g^{-1}) - (\zeta + \zeta^{-1})\text{id})$ has signature $(2, *)$ , where $\zeta$ is a primitive $p$ -th root of unity. This ensures $g$ (or $g^{-1}$ ) is a distinguished generator of the cyclic group.
Algorithmic Execution: The authors apply Algorithms 1, 2, 3, and 8 from [1] (implemented in OSCAR) to determine the isometry classes of the invariant and coinvariant lattices.
Verification and Correction: The authors systematically re-examine the pairs $(L^G, L_G)$ listed in the original [3, Tables 1–5]. They identify missing orthogonal complements, detect duplicates arising from lattice isometries (specifically $[2] \oplus [-4] \cong [-2] \oplus [4]$ ), and verify the existence of specific involutions.

Key Contributions and Results
The primary result of this note is the correction and completion of the classification, establishing that there are exactly 45 conjugacy classes of effective nonsymplectic groups $G \subset O(L)$ of prime order.

The specific corrections to the original tables include:

Missing Cases: The authors identified and added several missing pairs of invariant and coinvariant sublattices. Notable additions include:
- The pair $(\Lambda^G, \Lambda_G) = (U^{\oplus 3} \oplus [-2], U \oplus [2])$ in Table 1.
- New entries in Table 2 involving orthogonal complements such as $U^{\oplus 2} \oplus A_3(-1)$ .
- Missing orthogonal complements $T$ for specific sublattices $S$ in Table 3, such as $T = U(2) \oplus [-4]$ and $T = U \oplus A_3(-1)$ .
- Missing pairs in Table 5, including $(L^G, L_G) = (U \oplus [2] \oplus [-2]^{\oplus 3}, U(2))$ .
Removal of Duplicates: Entries in [3, Table 3] (specifically Nos. 9, 10, 15, 16, 19, and 20) were identified as duplicates of other entries due to the lattice isometry $[2] \oplus [-4] \cong [-2] \oplus [4]$ . These were removed in the corrected Table 3.
Correction of Errors: An erroneous entry in [3, Table 5] (No. 24) was identified and corrected. The original entry suggested a primitive extension that does not exist within the lattice $U^{\oplus 3} \oplus [-2]^{\oplus 2}$ .
Completeness of Involutions: The authors clarified which entries in Table 3 correspond to actual involutions of $L$ , marking specific missing cases with a trefoil symbol ("♣") to indicate that the associated coinvariant lattice arises from an involution.

Significance
The paper claims modestly that its purpose is to provide a "complete list of corrections" for the classification presented in [3]. By rectifying missing cases and removing duplicates, the authors ensure the lattice-theoretic classification of nonsymplectic automorphisms on OG6 manifolds is exhaustive. The corrected tables (1 through 5) now accurately reflect the isometry classes of invariant and coinvariant sublattices and the number of conjugacy classes of groups associated with them. This correction is essential for researchers relying on the classification of effective nonsymplectic isometries for further theoretical developments in the geometry of irreducible holomorphic symplectic manifolds.

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