# Collapsing hyperk\"ahler manifolds

## TL;DR

This paper proves that certain hyperk"ahler manifolds with Lagrangian fibrations collapse to a lower-dimensional space with special K"ahler geometry as fiber volumes shrink, revealing geometric and topological structure.

## Contribution

It establishes the Gromov-Hausdorff collapse of hyperk"ahler manifolds with shrinking fibers to a special K"ahler base, detailing the geometric limits and singularities.

## Key findings

- Hyperk"ahler metrics collapse to a special K"ahler manifold.
- Collapse occurs smoothly away from singular fibers.
- The limit space is homeomorphic to the base projective space.

## Abstract

Given a projective hyperkahler manifold with a holomorphic Lagrangian fibration, we prove that hyperkahler metrics with volume of the torus fibers shrinking to zero collapse in the Gromov-Hausdorff sense (and smoothly away from the singular fibers) to a compact metric space which is a half-dimensional special Kahler manifold outside a singular set of real Hausdorff codimension 2 and is homeomorphic to the base projective space.