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Compactifying Lagrangian fibrations

Giulia Saccà

TL;DR

The paper develops a general framework to compactify quasi-projective Lagrangian fibrations by holomorphic symplectic varieties, with a criterion guaranteeing a (possibly singular) Q-factorial terminal symplectic compactification extending the fibration over a normal base B. Central tools include extendable cycle-theoretic holomorphic 2-forms and a minimal-model-program approach to obtain nef canonical classes over B, plus a relative Albanese construction that produces a smooth abelian fibration A→B^{nm} acting on X. The author then shows that any almost torsor over A or over a group isogenous to A admits a symplectic compactification, and that if fibers are integral, these compactifications can be smooth and locally isomorphic to the original fibration; this yields new contexts for Shafarevich–Tate twists and links to OG6/OG10-type hyper-Kähler manifolds. The results unify several known examples (e.g., intermediate Jacobian fibrations) under a single compactification criterion and provide a robust framework for extending geometric Lagrangian fibrations while controlling deformation types and Hodge structures.

Abstract

We suggest a general framework for compactifing quasi-projective Lagrangian fibrations of geometric origin by holomorphic symplectic varieties. This framework includes a compactification criterion, which we then apply to various fibrations of geometric origin, and a discussion on holomorphic forms that are defined via correspondences in geometric examples. As application, we show that given a Lagrangian fibration $X \to B$ admitting local sections over an open subset $V$ with codimension $\ge 2$ complement, there exists a (possibly singular) holomorphic symplectic compactification of the Albanese fibration $A \to V$ (which we show exists as a smooth commutative algebraic group with connected fibers acting on $X_{V}$), as well as of any other torsor over $A$, or over any smooth commutative group scheme over $B$ with connected fibers that is isogenous to $A$.

Compactifying Lagrangian fibrations

TL;DR

The paper develops a general framework to compactify quasi-projective Lagrangian fibrations by holomorphic symplectic varieties, with a criterion guaranteeing a (possibly singular) Q-factorial terminal symplectic compactification extending the fibration over a normal base B. Central tools include extendable cycle-theoretic holomorphic 2-forms and a minimal-model-program approach to obtain nef canonical classes over B, plus a relative Albanese construction that produces a smooth abelian fibration A→B^{nm} acting on X. The author then shows that any almost torsor over A or over a group isogenous to A admits a symplectic compactification, and that if fibers are integral, these compactifications can be smooth and locally isomorphic to the original fibration; this yields new contexts for Shafarevich–Tate twists and links to OG6/OG10-type hyper-Kähler manifolds. The results unify several known examples (e.g., intermediate Jacobian fibrations) under a single compactification criterion and provide a robust framework for extending geometric Lagrangian fibrations while controlling deformation types and Hodge structures.

Abstract

We suggest a general framework for compactifing quasi-projective Lagrangian fibrations of geometric origin by holomorphic symplectic varieties. This framework includes a compactification criterion, which we then apply to various fibrations of geometric origin, and a discussion on holomorphic forms that are defined via correspondences in geometric examples. As application, we show that given a Lagrangian fibration admitting local sections over an open subset with codimension complement, there exists a (possibly singular) holomorphic symplectic compactification of the Albanese fibration (which we show exists as a smooth commutative algebraic group with connected fibers acting on ), as well as of any other torsor over , or over any smooth commutative group scheme over with connected fibers that is isogenous to .

Paper Structure

This paper contains 6 sections, 23 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1']}
  3. Cycle theoretic holomorphic symplectic form
  4. Applications I: intermediate Jacobian fibrations
  5. The relative Albanese
  6. Applications II: Torsors and isogenous fibrations

Key Result

Theorem 1

[=Theorem thm1'] Let $B$ be a normal quasi-projective variety and let $U \subset B$ be a non-empty open subset. Let $\pi_U: M_U \to U$ be a surjective morphism from a quasi-projective holomorphic symplectic manifold (or a $\mathbb{Q}$-factorial terminal symplectic variety) and suppose that $\pi_U$ i Then there exists a $\mathbb{Q}$-factorial terminal symplectic variety $M$ containing $M_U$ as an o

Theorems & Definitions (72)

  • Theorem 1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Example 2.6
  • Definition 3.1
  • ...and 62 more