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Irreducible symplectic varieties with a large second Betti number

Yuchen Liu, Zhiyu Liu, Chenyang Xu

TL;DR

The paper develops a general criterion for compactifying non-compact Lagrangian fibrations to irreducible symplectic varieties using MMP techniques and extendability of holomorphic 2-forms. It then applies this to the relative Jacobian fibration associated with cubic fivefolds containing a fixed cubic fourfold, constructing a projective base $M\cong \mathbb{P}(1^{15},2^6,3)$ via VGIT and producing a $\{Q}$-factorial terminal irreducible symplectic compactification with a Lagrangian fibration over that base, achieving $b_2\ge 24$. This yields a genuinely new deformation type of irreducible symplectic varieties not deformation equivalent to previously known examples, and provides a concrete link between Lagrangian fibrations, base compactifications, and moduli via VGIT. The results support Markman’s rank-1 obstruction paradigm in this setting and broaden the landscape of higher-$b_2$ irreducible symplectic geometries through explicit constructions.

Abstract

We prove a general result on the existence of irreducible symplectic compactifications of non-compact Lagrangian fibrations. As an application, we show that the relative Jacobian fibration of cubic fivefolds containing a fixed cubic fourfold can be compactified by a $\mathbb{Q}$-factorial terminal irreducible symplectic variety with the second Betti number at least 24, and admits a Lagrangian fibration whose base is a weighted projective space. In particular, it belongs to a new deformation type of irreducible symplectic varieties.

Irreducible symplectic varieties with a large second Betti number

TL;DR

The paper develops a general criterion for compactifying non-compact Lagrangian fibrations to irreducible symplectic varieties using MMP techniques and extendability of holomorphic 2-forms. It then applies this to the relative Jacobian fibration associated with cubic fivefolds containing a fixed cubic fourfold, constructing a projective base via VGIT and producing a -factorial terminal irreducible symplectic compactification with a Lagrangian fibration over that base, achieving . This yields a genuinely new deformation type of irreducible symplectic varieties not deformation equivalent to previously known examples, and provides a concrete link between Lagrangian fibrations, base compactifications, and moduli via VGIT. The results support Markman’s rank-1 obstruction paradigm in this setting and broaden the landscape of higher- irreducible symplectic geometries through explicit constructions.

Abstract

We prove a general result on the existence of irreducible symplectic compactifications of non-compact Lagrangian fibrations. As an application, we show that the relative Jacobian fibration of cubic fivefolds containing a fixed cubic fourfold can be compactified by a -factorial terminal irreducible symplectic variety with the second Betti number at least 24, and admits a Lagrangian fibration whose base is a weighted projective space. In particular, it belongs to a new deformation type of irreducible symplectic varieties.
Paper Structure (15 sections, 31 theorems, 110 equations)

This paper contains 15 sections, 31 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Existence
  3. A new deformation type of irreducible symplectic varieties
  4. Preliminaries
  5. Symplectic varieties
  6. Decomposition theorem
  7. Lagrangian fibrations
  8. Existence of compactifications
  9. Symplectic compactifications
  10. Irreducible symplectic compactifications
  11. Variant
  12. Irreducible symplectic varieties with $b_2\geq 24$
  13. Variation of GIT
  14. The moduli space of cubic fivefolds containing a fixed cubic fourfold
  15. The Jacobian fibration

Key Result

Theorem 1.1

Let $(X_0,\sigma_0)$ be a holomorphic symplectic manifold with a dominant morphism to a normal projective variety $B$. Assume that $\sigma_0$ extends to a holomorphic $2$-form on a smooth compactification of $\pi_0$. If then there exists an irreducible symplectic variety $X$ with $\mathbb{Q}$-factorial terminal singularities and a Lagrangian fibration $\pi\colon X\rightarrow B$ extending $\pi_0$

Theorems & Definitions (64)

  • Theorem 1.1: Theorem \ref{['thm-irr-sym']}
  • Theorem 1.2: Theorem \ref{['thm-cubic']}
  • Remark 1.3
  • Conjecture 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 54 more