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The Néron model of a higher-dimensional Lagrangian fibration

Yoon-Joo Kim

TL;DR

This work extends Néron model theory to higher-dimensional Lagrangian fibrations by constructing two main objects: a Néron model $X^{n}\to B$ of the smooth locus fibration and a Néron model $P\to B$ of the automorphism abelian scheme, with $X^{n}$ forming a $P$-torsor. Key innovations include the development of a robust theory of group algebraic spaces, a $\oldsymbol{\delta}$-regular action on the fibration, and a detailed analysis of codimension-1 and deeper $\,\delta$-strata behavior via birational transformations and flops. The paper proves the existence of these Néron models, describes their structure as unions of smooth loci from birational models, and explores when the smooth locus $X'$ is a torsor under $P$ (or of finite type/separated) and when it fails, providing several explicit examples. The results illuminate the relationship between birational geometry and Néron-type compactifications, connect to dual Picard data, and have implications for the holomorphic Strominger–Yau–Zaslow program in higher dimensions. Overall, the framework offers a canonical, though sometimes non-separated or non-finite-type, compactification capturing birational automorphisms that act as translations on generic abelian fibers.

Abstract

Let $π: X \to B$ be a projective Lagrangian fibration of a smooth symplectic variety $X$ to a smooth variety $B$. Denote the complement of the discriminant locus by $B_0 = B \setminus \operatorname{Disc}(π)$, its preimage by $X_0 = π^{-1}(B_0)$, and the complement of the critical locus by $X' = X \setminus \operatorname{Sing}(π)$. Under an assumption that the morphism $X' \to B$ is surjective, we construct (1) the Néron model of the abelian fibration $π_0 : X_0 \to B_0$ and (2) the Néron model of its automorphism abelian scheme $\operatorname{Aut}^{\circ}_{π_0} \to B_0$. Contrary to the case of elliptic fibrations, $X'$ may not be the Néron model of $X_0$; this is precisely because of the existence of flops in higher-dimensional symplectic varieties. Using such techniques, we analyze when $X' \to B$ is a torsor under a smooth group scheme and also revisit some known results in the literature.

The Néron model of a higher-dimensional Lagrangian fibration

TL;DR

This work extends Néron model theory to higher-dimensional Lagrangian fibrations by constructing two main objects: a Néron model of the smooth locus fibration and a Néron model of the automorphism abelian scheme, with forming a -torsor. Key innovations include the development of a robust theory of group algebraic spaces, a -regular action on the fibration, and a detailed analysis of codimension-1 and deeper -strata behavior via birational transformations and flops. The paper proves the existence of these Néron models, describes their structure as unions of smooth loci from birational models, and explores when the smooth locus is a torsor under (or of finite type/separated) and when it fails, providing several explicit examples. The results illuminate the relationship between birational geometry and Néron-type compactifications, connect to dual Picard data, and have implications for the holomorphic Strominger–Yau–Zaslow program in higher dimensions. Overall, the framework offers a canonical, though sometimes non-separated or non-finite-type, compactification capturing birational automorphisms that act as translations on generic abelian fibers.

Abstract

Let be a projective Lagrangian fibration of a smooth symplectic variety to a smooth variety . Denote the complement of the discriminant locus by , its preimage by , and the complement of the critical locus by . Under an assumption that the morphism is surjective, we construct (1) the Néron model of the abelian fibration and (2) the Néron model of its automorphism abelian scheme . Contrary to the case of elliptic fibrations, may not be the Néron model of ; this is precisely because of the existence of flops in higher-dimensional symplectic varieties. Using such techniques, we analyze when is a torsor under a smooth group scheme and also revisit some known results in the literature.

Paper Structure

This paper contains 59 sections, 127 theorems, 208 equations, 1 table.

Table of Contents

  1. Introduction
  2. Relation to results in literature and applications
  3. Sketch proofs and organization of the paper
  4. Group algebraic spaces
  5. Lie algebra spaces
  6. Deformation to the Lie algebra
  7. Neutral components
  8. Group spaces with connected fibers and strict neutral component
  9. Connected group spaces and (weak) neutral component
  10. Alignment
  11. Étale group algebraic spaces
  12. Main component
  13. Automorphism space, Picard space, and examples
  14. The automorphism group space
  15. The Picard group space
  16. ...and 44 more sections

Key Result

theorem 1.2

Let $\pi : X \longrightarrow B$ be a projective Lagrangian fibration from a smooth symplectic variety $X$ to a smooth variety $B$, both defined over $\mathbb{C}$. Let $B_0 \subset B$ be the Zariski open subset over which $\pi$ is smooth and $\pi_0 : X_0 = \pi^{-1}(B_0) \longrightarrow B_0$ a smooth As a result, $X^n$ is a $P$-torsor.

Theorems & Definitions (289)

  • definition 1.1
  • theorem 1.2: Existence of the Néron models
  • theorem 1.3: Structure of the Néron model $P$
  • remark 1.4
  • theorem 1.6: Structure of the Néron model $X^n$
  • proposition 1: Failure of separatedness of $X^n$ and the torsor property of $X'$
  • theorem 1.7: Dual Néron model
  • definition 2.1
  • theorem 2.2: Cartier's theorem
  • proposition 2
  • ...and 279 more