Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lagrangian fibrations on Nikulin-type orbifolds

Giacomo Nanni

TL;DR

The paper achieves a complete classification of Lagrangian fibrations on Nikulin-type orbifolds by reducing to two deformation families corresponding to fibrations on a K3^[2]-type fourfold with a symplectic involution, either invariant (type A) or anti-invariant (type B). It combines explicit geometric constructions (Fermat quartic and Markushevich–Tikhomirov Prym), lattice-theoretic monodromy analysis, and a detailed study of isotropic nef classes to show that MT fibrations are type A and that exactly two fibration classes exist up to monodromy. The results yield two deformation families and establish that the SYZ conjecture holds for Nikulin-type orbifolds, with a corollary of Kobayashi pseudometric vanishing on these spaces. Overall, the work extends the SYZ framework to singular irreducible holomorphic symplectic varieties and clarifies how symmetry under symplectic involutions governs the fibration geometry.

Abstract

We classify lagrangian fibrations on Nikulin orbifolds, a well studied class of singular irreducible holomorphic symplectic varieties, and prove they verify the SYZ conjecture.

Lagrangian fibrations on Nikulin-type orbifolds

TL;DR

The paper achieves a complete classification of Lagrangian fibrations on Nikulin-type orbifolds by reducing to two deformation families corresponding to fibrations on a K3^[2]-type fourfold with a symplectic involution, either invariant (type A) or anti-invariant (type B). It combines explicit geometric constructions (Fermat quartic and Markushevich–Tikhomirov Prym), lattice-theoretic monodromy analysis, and a detailed study of isotropic nef classes to show that MT fibrations are type A and that exactly two fibration classes exist up to monodromy. The results yield two deformation families and establish that the SYZ conjecture holds for Nikulin-type orbifolds, with a corollary of Kobayashi pseudometric vanishing on these spaces. Overall, the work extends the SYZ framework to singular irreducible holomorphic symplectic varieties and clarifies how symmetry under symplectic involutions governs the fibration geometry.

Abstract

We classify lagrangian fibrations on Nikulin orbifolds, a well studied class of singular irreducible holomorphic symplectic varieties, and prove they verify the SYZ conjecture.
Paper Structure (14 sections, 16 theorems, 47 equations)

This paper contains 14 sections, 16 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Examples of fibrations
  5. On the Fermat quartic
  6. The Markushevich-Tikhomirov Prym
  7. Monodromy
  8. Isotropic classes
  9. Classification of fibrations
  10. Equivariant fibrations on the smooth fourfold
  11. Trivial action on the base: type A
  12. Non-trivial action on the base: type B
  13. The Markushevich-Tikhomirov system
  14. Consequences of the classification

Key Result

Theorem 1.3

There are 2 deformation families of lagrangian fibrations on Nikulin-type orbifolds. In particular, let $Y$ be a Nikulin-type orbifold and $\phi: Y\rightarrow B$ a lagrangian fibration, then $\phi$ can be deformed to one of the following examples:

Theorems & Definitions (29)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3: \ref{['thm:classificationFinal']}
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Corollary 2.7
  • ...and 19 more