[Paper Review] Symplectic involutions of $K3^{[n]}$ type and Kummer $n$ type manifolds

Ljudmila Kamenova, Giovanni Mongardi+1 more|arXiv (Cornell University)|Sep 8, 2018

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper classifies the fixed loci of symplectic involutions on hyperkähler manifolds of $K3^{[n]}$ type and Kummer $n$ type, proving they consist of finitely many deformations of Hilbert schemes of $K3$ surfaces and isolated points. The key result is a precise dimension-wise decomposition of the fixed locus, with the largest component being a single deformation of $S^{[n/2]}$ (or eight copies if $n$ is odd), derived via moduli space connectivity and global Torelli theorems.

ABSTRACT

In this paper we describe the fixed locus of a symplectic involution on a hyperkähler manifold of type $K3^{[n]}$ or of Kummer $n$ type. We prove that the fixed locus consists of finitely many copies of Hilbert schemes of $K3$ surfaces of lower dimensions and isolated fixed points.

Motivation & Objective

• To classify the fixed loci of symplectic involutions on hyperkähler manifolds of $K3^{[n]}$ type and Kummer $n$ type.
• To generalize Nikulin’s result on $K3$ surfaces (8 fixed points) and Mongardi’s result on $K3^{[2]}$ (28 points and one $K3$) to arbitrary $n$.
• To prove that the fixed locus decomposes into finitely many components, each a deformation of a Hilbert scheme $S^{[m]}$ for $m \leq n/2$ (or $m \leq (n+1)/2$ in the Kummer case), and isolated points when $n \leq 24$ or $n \leq 48$.
• To establish that the moduli space of such pairs $(X, \iota)$ is connected, enabling reduction to standard geometric models.

Proposed method

• Use the Global Torelli theorem to show that any pair $(X, \iota)$ with $X$ of $K3^{[n]}$ or Kummer $n$ type and $\iota$ a symplectic involution is birational to a 'standard pair' arising from a $K3$ surface or abelian surface with an involution.
• Prove that numerically standard group actions on cohomology imply the pair is a standard pair, allowing reduction to known geometric constructions.
• Leverage the fact that the moduli space of pairs $(X, \iota)$ is connected to deform any such pair to a standard one while preserving the group action.
• Analyze the fixed locus via the induced action on $\mathrm{Sym}^n S$ and $\mathrm{Sym}^n A$, using combinatorics to count components based on parity of $n$.
• For $K3^{[n]}$ type, use quiver variety techniques and root systems of the $\hat{\mathfrak{sl}}_2$ algebra to describe the fixed locus as a union of quiver varieties $\mathfrak{M}_{\mathbf{Q}'_0}((n_1,n_2),(0,1))$, with equality in the even case and at least two components in the odd case.
• For Kummer $n$ type, reduce to the sign involution on the abelian surface $A$, and use similar techniques to classify the fixed locus in terms of $S^{[m]}$ deformations and isolated points.

Experimental results

Research questions

• RQ1 What is the structure of the fixed locus of a symplectic involution on a hyperkähler manifold of $K3^{[n]}$ type?
• RQ2 How does the fixed locus decompose into components of different dimensions, and what determines the number of connected components in each dimension?
• RQ3 Can the fixed locus be described uniformly across all $n$, and does it depend on the parity of $n$?
• RQ4 Is the moduli space of such pairs $(X, \iota)$ connected, enabling deformation to a standard geometric model?
• RQ5 What is the role of the cohomological action in determining whether a group action is numerically standard?

Key findings

• The fixed locus of a symplectic involution on a $K3^{[n]}$ type manifold consists of finitely many components, each a deformation of a Hilbert scheme $S^{[m]}$ for $m \leq n/2$, and possibly isolated fixed points when $n \leq 24$.
• The fixed locus is stratified by even-dimensional components $F_{2m}$ with $\max(0, n/2 - 12) \leq m \leq n/2$, and each $F_{2m}$ has $\left| \sum_{n-k-2l=2m} \binom{8}{k} \binom{k}{l} \right|$ connected components.
• When $n$ is even, the largest-dimensional fixed component is a single deformation of $S^{[n/2]}$; when $n$ is odd, it consists of eight such components.
• For Kummer $n$ type manifolds, the fixed locus similarly decomposes into $F_{2m}$ with $\max(0, (n+1)/2 - 24) \leq m \leq (n+1)/2$, and each $F_{2m}$ has $N^n_m$ components, each a deformation of $S^{[m]}$, with isolated points possible only if $n \leq 48$.
• The proof relies on showing that the moduli space of pairs $(X, \iota)$ is connected, allowing deformation to a standard pair, and that numerically standard actions on cohomology imply the pair is standard.
• In the case $n=2$, the fixed locus is isomorphic to a quiver variety $\mathfrak{M}_{\mathbf{Q}'_0}((1,1),(0,1))$, and for $n=3$, it has two components: one isomorphic to $\mathfrak{M}_{\mathbf{Q}'_0}((2,1),(0,1))$ and one isolated point, corresponding to the punctual Hilbert scheme at the origin.