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Symplectic actions of groups of order 4 on K3^[2]-type manifolds, and standard involutions on Nikulin-type orbifolds

Benedetta Piroddi

TL;DR

This work analyzes symplectic actions of order-4 groups on $K3^{[2]}$-type manifolds and the resulting Nikulin-type orbifolds obtained by terminalizing quotients by a symplectic involution. It develops a lattice-theoretic framework to classify projective $K3^{[2]}$-type manifolds with a symplectic $G$-action (where $|G|=4$), and then studies the induced involutions on the associated Nikulin orbifolds, detailing their action on $H^2(Y,\mathbb{Z})$ and fixed loci. The paper provides explicit projective models for the general members of the two main families (a Fano model over a cubic fourfold and a Hilbert square of a quartic surface with a mixed $G$-action) and gives a comprehensive lattice-based classification of Nikulin orbifolds admitting standard involutions, as well as those with mixed $(\\mathbb{Z}/2\\mathbb{Z})^2$ actions. The results illuminate how deformation, monodromy, and gluing of invariant and co-invariant lattices govern the existence and form of induced involutions on Nikulin-type orbifolds, with broader implications for the study of quotient varieties in hyperkähler geometry.

Abstract

Given a K3^[2]-type manifold X with a symplectic involution i, the quotient X/i admits a Nikulin orbifold Y as terminalization. We study the symplectic action of a group G of order 4 on X, such that i belongs to G, and the natural involution induced on Y (the two groups give two different results). We give a lattice-theoretic classification of X and Y in the projective case, and give some explicit examples of models of X. We also give lattice-theoretic criteria that a Nikulin-type orbifold N has to satisfy to admit a symplectic involution that deforms to an induced one.

Symplectic actions of groups of order 4 on K3^[2]-type manifolds, and standard involutions on Nikulin-type orbifolds

TL;DR

This work analyzes symplectic actions of order-4 groups on -type manifolds and the resulting Nikulin-type orbifolds obtained by terminalizing quotients by a symplectic involution. It develops a lattice-theoretic framework to classify projective -type manifolds with a symplectic -action (where ), and then studies the induced involutions on the associated Nikulin orbifolds, detailing their action on and fixed loci. The paper provides explicit projective models for the general members of the two main families (a Fano model over a cubic fourfold and a Hilbert square of a quartic surface with a mixed -action) and gives a comprehensive lattice-based classification of Nikulin orbifolds admitting standard involutions, as well as those with mixed actions. The results illuminate how deformation, monodromy, and gluing of invariant and co-invariant lattices govern the existence and form of induced involutions on Nikulin-type orbifolds, with broader implications for the study of quotient varieties in hyperkähler geometry.

Abstract

Given a K3^[2]-type manifold X with a symplectic involution i, the quotient X/i admits a Nikulin orbifold Y as terminalization. We study the symplectic action of a group G of order 4 on X, such that i belongs to G, and the natural involution induced on Y (the two groups give two different results). We give a lattice-theoretic classification of X and Y in the projective case, and give some explicit examples of models of X. We also give lattice-theoretic criteria that a Nikulin-type orbifold N has to satisfy to admit a symplectic involution that deforms to an induced one.
Paper Structure (14 sections, 23 theorems, 35 equations, 2 tables)

This paper contains 14 sections, 23 theorems, 35 equations, 2 tables.

Table of Contents

  1. Projective families of K3$^{[2]}$-type manifolds with a symplectic action of a group of order 4
  2. Families of projective K3$^{[2]}$-type manifolds with a symplectic action of $\mathbb Z/4\mathbb Z$
  3. Families of projective K3$^{[2]}$-type manifolds with a symplectic action of $(\mathbb Z/2\mathbb Z)^2$
  4. Fixed loci
  5. Projective examples
  6. The induced involutions on Nikulin orbifolds
  7. Nikulin orbifolds and the cohomology of quotients
  8. Natural and standard actions
  9. The case $G=\mathbb Z/4\mathbb Z$
  10. The case $G=(\mathbb Z/2\mathbb Z)^2$
  11. Induced involutions on Nikulin orbifolds
  12. Classifying Nikulin-type Orbifolds with a standard symplectic involution
  13. Families of Nikulin orbifolds with an induced symplectic involution
  14. Nikulin-type orbifolds with a symplectic action of $(\mathbb Z/2\mathbb Z)^2$

Key Result

Proposition 1.0.4

A given symplectic action of a finite group $G$ admits at most countably many projective families. A projective hyperkähler manifold $X$ admits the chosen action of $G$ if and only if it belongs to one such projective family.

Theorems & Definitions (64)

  • Definition 1.0.1: see Dolgachev
  • Remark 1.0.2
  • Definition 1.0.3
  • Proposition 1.0.4
  • proof
  • Remark 1.0.5
  • Remark 1.0.6
  • Remark 1.0.7
  • Theorem 1.1.1
  • proof
  • ...and 54 more