Fixed loci of symplectic automorphisms of $K3^{[n]}$ and $n$-Kummer type manifolds

TL;DR

This work extends the description of fixed loci for finite symplectic group actions from involutions to general groups on hyperkähler manifolds of $K3^{[n]}$ and $n$-Kummer type. By leveraging the BryanGyenge framework and deformation theory, the authors prove that all irreducible components of the fixed loci are deformation equivalent to $K3^{[k]}$-type manifolds (or isolated points), with precise counts given by theta-series associated to invariant lattices $M^G$. In the Hilbert-scheme case, component counts are $N_k = \Theta_G[n-|G|k]$, while for generalized Kummer varieties, regular groups yield $N_k = \Theta_G[n-|G|k;1]$ after refining by the Albanese-fiber data. The results require numerically standard (and in some cases standard) group actions and provide explicit generating functions for enumerating fixed components, unifying the K3 and Kummer settings and correcting previous lower-dimensional component assertions. This yields a practical, lattice-theory-driven classification of fixed loci with potential applications to moduli and mirror-symmetry studies of hyperkähler manifolds.

Abstract

The aim of this paper is to give an explicit description of the fixed loci of symplectic automorphisms for certain hyperkahler manifolds, namely for Hilbert schemes on K3 surfaces and for generalized Kummer varieties. Here we extend our previous results from the case of involutions to more general groups. In particular, under some conditions on the dimension, we give the full answer for finite group actions of symplectic automorphisms coming from K3 surfaces. We prove that the all irreducible components of the fixed loci are of $K3^{[k]}$ type of lower dimensions or isolated points.

Theorems & Definitions (37)

• Theorem 1.0.1
• Theorem 1.0.2
• Theorem 1.0.3
• Theorem 1.0.4
• Definition 2.0.1
• Remark 2.0.2
• Definition 2.0.3
• Definition 2.0.4
• Remark 2.0.5
• Theorem 2.0.6