[Paper Review] Addendum to: On fibre space structures of a projective irreducible symplectic manifold

Daisuke Matsushita|ArXiv.org|Mar 8, 1999

Geometry and complex manifolds2 references14 citations

TL;DR

This paper proves that every fibration from a projective irreducible symplectic manifold to a projective base is a Lagrangian fibration, meaning general fibers are Lagrangian submanifolds with respect to the holomorphic symplectic form. Using a bilinear form on cohomology and comparison of cohomological terms, the authors show the vanishing of a key integral, confirming the Lagrangian condition and implying fibers are abelian varieties.

ABSTRACT

In this note, we prove that every fibre space structures of a projective irreducible symplectic manifold is a lagrangian fibration.

Motivation & Objective

• To establish that every fibration of a projective irreducible symplectic manifold over a projective base is a Lagrangian fibration.
• To resolve a gap in understanding whether non-Lagrangian fibrations can exist in this class of manifolds.
• To show that the general fiber of such a fibration must be an abelian variety, as a consequence of the Lagrangian condition.
• To clarify the distinction between families of non-Lagrangian tori and actual fibrations, as noted in prior literature.

Proposed method

• Use of the holomorphic symplectic form ω and its conjugate ω̄ on a 2n-dimensional projective irreducible symplectic manifold X.
• Application of the bilinear form q_X on H^2(X, ℂ) as defined in [3, Theorem 4.7], which satisfies q_X(D,D)^n = D^{2n} for D ∈ H^2(X, ℂ).
• Construction of a family of cohomology classes ω + ω̄ + sA + tH for parameters s, t, and comparison of the 2n-th power of this class on both sides of the identity.
• Identification and cancellation of cross-terms involving q_X(ω + ω̄, A) and q_X(ω + ω̄, H), which vanish due to Hodge-theoretic orthogonality from [1, Théorème 5].
• Comparison of the coefficient of s^{n-2}t^n in the expansion of both sides to deduce ω̄ ∧ ω ∧ A^{n-2} ∧ H^n = 0.
• Use of this vanishing to conclude that ∫_F ω ∧ ω̄ ∧ A^{n-2} = 0, confirming the Lagrangian condition on the fiber F.

Experimental results

Research questions

• RQ1 Is every fibration from a projective irreducible symplectic manifold to a projective base necessarily a Lagrangian fibration?
• RQ2 Can a family of non-Lagrangian tori in an irreducible symplectic manifold fail to form a fibration, as suggested by Markshevich?
• RQ3 Does the vanishing of the integral ∫_F ω ∧ ω̄ ∧ A^{n-2} imply that the general fiber F is Lagrangian?
• RQ4 What is the role of the bilinear form q_X in characterizing Lagrangian fibrations in this context?
• RQ5 How does the cohomological structure of the manifold constrain the geometry of its fibrations?

Key findings

• Every fibration f: X → B from a projective irreducible symplectic manifold X to a projective base B is a Lagrangian fibration.
• The general fiber F of such a fibration is a Lagrangian submanifold, meaning the restriction of the symplectic form ω to F vanishes identically.
• The vanishing of the integral ∫_F ω ∧ ω̄ ∧ A^{n-2} is established via cohomological comparison, confirming the Lagrangian condition.
• The bilinear form q_X ensures that cross-terms involving ω + ω̄ and A or H vanish, which is essential for the proof.
• The result implies that the general fiber is an abelian variety, as Lagrangian fibrations are complete integrable systems by Beauville’s result.
• The proof resolves a potential confusion in the literature by showing that non-Lagrangian tori cannot form fibrations, even if they exist as families.