Special K\"ahler geometry and holomorphic Lagrangian fibrations

TL;DR

This paper explores the differential geometry of special K"ahler metrics in holomorphic Lagrangian fibrations of hyperk"ahler manifolds, leading to a new proof that the base space must be projective space.

Contribution

It introduces a novel geometric approach to prove that the base of such fibrations is necessarily projective space, extending previous results by integrating special K"ahler geometry and rational curves.

Findings

The pullback of the tangent bundle admits a parallel splitting.

The base space must be projective space.

The approach generalizes previous proofs of the base being projective space.

Abstract

Given a holomorphic Lagrangian fibration of a compact hyperkahler manifold, we use the differential geometry of the special Kahler metric that exists on the base away from the discriminant locus, and show that the pullback of the tangent bundle of the base to the total space of a family of minimal rational curves admits a parallel splitting. The splitting is nontrivial when the base is not half-dimensional projective space. Combining this with results of Voisin, Hwang and Bakker-Schnell, we deduce that the base must be projective space, a result first proved by Hwang.