A Proof of Mongardi's Conjecture on Finite Symplectic Group Actions on Hyperk\"ahler Manifolds of $K3^{[n]}$ Type

Abstract

Let a finite group act faithfully by symplectic automorphisms on a manifold of $K3^{[n]}$ type for $n\geq2$. We prove Mongardi's conjecture that the strict inequality $\mathrm{rk}(S)+\ell(A_S)<24$ holds for its associated Leech coinvariant lattice $S$. The converse was already proved independently by Huybrechts and Mongardi: any Leech coinvariant lattice $S$ satisfying the strict inequality and of rank at most $20$ is realized by such an action for some $n\geq2$. The key idea is to consider the minimum norms of vectors representing discriminant classes of $S$. The absence of numerical walls gives lower bounds for certain classes, while the structure of Leech lattice implies upper bounds for all discriminant classes. Comparing these bounds reduces the proof to finitely many values of $n$. We exclude the remaining cases using further lattice arguments, together with H\"ohn--Mason's classification of fixed-point sublattices of Leech lattice, thereby proving Mongardi's conjecture.