[Paper Review] A canonical bundle formular of projective Lagrangian fibrations

Daisuke Matsushita|arXiv (Cornell University)|Sep 30, 2007

Geometry and complex manifolds8 references3 citations

TL;DR

This paper classifies singular fibers of projective Lagrangian fibrations over codimension one loci, leading to a canonical bundle formula for such fibrations over smooth base manifolds. The approach combines algebro-geometric techniques with symplectic geometry to derive a global formula for the canonical bundle in terms of the fibration's geometry and singular fiber types.

ABSTRACT

We classify singular fibres of a projective Lagrangian fibration over codimension one points. As an application, we obtain a canonical bundle formula for a projective Lagrangian fibration over a smooth manifold.

Motivation & Objective

• To classify the structure of singular fibers in projective Lagrangian fibrations over codimension one points in the base.
• To understand the geometric and cohomological properties of these singular fibers in the context of holomorphic symplectic geometry.
• To derive a global canonical bundle formula for projective Lagrangian fibrations over smooth manifolds using the classification of singular fibers.
• To extend known results on canonical bundles in fibrations to the setting of Lagrangian fibrations with singular fibers.

Proposed method

• Utilizes the theory of Lagrangian fibrations in holomorphic symplectic geometry to analyze fibers over codimension one subvarieties.
• Applies techniques from birational geometry and minimal model program to classify possible singular fiber types.
• Employs the relative canonical bundle formula from algebraic geometry, adapted to the symplectic setting.
• Relies on the existence of a holomorphic symplectic form on the total space to constrain the structure of singular fibers.
• Uses the fact that the fibration is projective to apply standard tools from algebraic geometry, such as the relative canonical sheaf and multiplier ideal sheaves.
• Combines local analysis near singular fibers with global considerations to derive a formula for the canonical bundle of the total space.

Experimental results

Research questions

• RQ1 What are the possible types of singular fibers that can occur in a projective Lagrangian fibration over a codimension one locus?
• RQ2 How do the singular fibers affect the canonical bundle of the total space of the fibration?
• RQ3 Can a canonical bundle formula be derived for projective Lagrangian fibrations over smooth bases, analogous to known formulas in other fibration types?
• RQ4 What constraints does the holomorphic symplectic structure impose on the geometry of singular fibers?

Key findings

• The paper classifies the possible singular fiber types over codimension one points in the base, showing they are constrained by the symplectic structure.
• It establishes a canonical bundle formula for the total space of the fibration in terms of the base canonical bundle and the singular fiber contributions.
• The formula generalizes the classical canonical bundle formula in the context of Lagrangian fibrations with singular fibers.
• The result holds under the assumption that the fibration is projective and the base is smooth, ensuring the applicability of standard algebraic geometry tools.
• The singular fiber types are shown to contribute in a controlled way to the canonical bundle, consistent with the symplectic nature of the fibration.
• The derived formula provides a global expression for the canonical bundle, reflecting both the smoothness of the base and the geometry of the singular fibers.