Characteristic foliation on the discriminantal hypersurface of a holomorphic Lagrangian fibration

TL;DR

This paper classifies singular fibers of holomorphic Lagrangian fibrations using characteristic vector fields, revealing that their characteristic foliations have algebraic leaves that are rational or elliptic curves.

Contribution

It provides a Kodaira-type classification for singular fibers of holomorphic Lagrangian fibrations in Fujiki's class, based on characteristic vector fields.

Findings

Characteristic foliation leaves are algebraic, either rational or elliptic curves.

The approach is based on the symplectic form and the defining equation of the discriminantal hypersurface.

The classification extends Kodaira's work to a broader class of fibrations.

Abstract

We give a Kodaira-type classification of general singular fibers of a holomorphic Lagrangian fibration in Fujiki's class $\mathcal C$. Our approach is based on the study of the characteristic vector field of the discriminantal hypersurface, which naturally arises from the defining equation of the hypersurface via the symplectic form. As an application, we show that the characteristic foliation of the discriminantal hypersurface has algebraic leaves which are either rational curves or smooth elliptic curves.