关于K3^{[n]}型超凯勒流形上有限辛群作用的蒙加迪猜想的一个证明
A Proof of Mongardi's Conjecture on Finite Symplectic Group Actions on Hyperkähler Manifolds of $K3^{[n]}$ Type
AI总结：
本文针对n≥2的K3^{[n]}型超凯勒流形上的有限辛群作用，通过分析李赫余不变量格的判别式类范数，结合格论证与李赫格不动点子格分类，证明了蒙加迪猜想。
AI中文摘要：
设有限群以辛自同构的方式忠实地作用于n≥2的K3^{[n]}型流形上。我们证明了蒙加迪猜想：对于其对应的李赫余不变量格S，严格不等式rk(S)+ℓ(A_S)<24成立。其逆命题已由惠布雷斯和蒙加迪分别独立证明：任何满足该严格不等式且秩不超过20的李赫余不变量格S，都可由某个n≥2的此类作用实现。核心思路是考虑表示S的判别式类的向量的最小范数。数值壁的缺失为某些类给出下界，而李赫格的结构为所有判别式类给出上界。比较这些界将证明简化为有限多个n值。我们结合进一步的格论证以及赫恩-梅森关于李赫格不动点子格的分类，排除了剩余情形，从而证明了蒙加迪猜想。
英文摘要：
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 Höhn--Mason's classification of fixed-point sublattices of Leech lattice, thereby proving Mongardi's conjecture.
