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Monodromy of Nikulin orbifolds

Giacomo Nanni

TL;DR

This work provides a geometry-driven proof that the monodromy group of a Nikulin orbifold $Y$ equals the full orientation-preserving isometry group of its BBF lattice, $Mon^2(Y)=O^+(H^2(Y,\mathbb{Z}))$. Building on the monodromy framework for $K3^{[2]}$-type fourfolds and a lattice-theoretic link via twisted unimodular sums, the authors derive the result by extending monodromy operators from the invariant lattice and exploiting Nikulin’s extension criterion. The approach avoids heavy lattice-generation arguments by leveraging the corresponding $K3^{[2]}$-type case and elementary lattice theory, clarifying the geometric content of monodromy in this setting. The findings reinforce the deep connection between automorphisms of the BBF lattice and deformation-invariant monodromy phenomena for symplectic orbifolds arising from symplectic involutions.

Abstract

We give a new proof for the maximality of the monodromy group of a Nikulin orbifold, a symplectic orbifold arising as terminalisation of a symplectic quotient of a $K3^{[2]}$-type fourfold.

Monodromy of Nikulin orbifolds

TL;DR

This work provides a geometry-driven proof that the monodromy group of a Nikulin orbifold equals the full orientation-preserving isometry group of its BBF lattice, . Building on the monodromy framework for -type fourfolds and a lattice-theoretic link via twisted unimodular sums, the authors derive the result by extending monodromy operators from the invariant lattice and exploiting Nikulin’s extension criterion. The approach avoids heavy lattice-generation arguments by leveraging the corresponding -type case and elementary lattice theory, clarifying the geometric content of monodromy in this setting. The findings reinforce the deep connection between automorphisms of the BBF lattice and deformation-invariant monodromy phenomena for symplectic orbifolds arising from symplectic involutions.

Abstract

We give a new proof for the maximality of the monodromy group of a Nikulin orbifold, a symplectic orbifold arising as terminalisation of a symplectic quotient of a -type fourfold.

Paper Structure

This paper contains 9 sections, 7 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattice notation
  4. Monodromy
  5. Sum of twisted unimodular lattices
  6. Isometries of the invariant lattice
  7. Results
  8. Preservation of $\Sigma_Y$ under isometries
  9. Proof of the main result

Key Result

Theorem 1.1

Let $Y$ be a Nikulin orbifold. Then $Mon^2(Y)=O^+(Y)$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Remark
  • Proposition 2.1
  • Remark
  • Proposition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Proposition 2.5
  • Remark
  • Corollary 2.6
  • ...and 3 more