Collapsing of negative K\"ahler-Einstein metrics

TL;DR

This paper investigates how negative K"ahler-Einstein metrics behave under degenerations of canonical polarized manifolds, showing they collapse to lower-dimensional affine K"ahler manifolds in a precise geometric sense.

Contribution

It proves that under certain degenerations, these metrics collapse to lower-dimensional affine K"ahler manifolds, revealing the geometric limits of such degenerations.

Findings

Metrics collapse to lower-dimensional manifolds

Limit spaces are real affine K"ahler manifolds

Collapse occurs in the Gromov-Hausdorff sense

Abstract

In this paper, we study the collapsing behaviour of negative K\"{a}hler-Einstein metrics along degenerations of canonical polarized manifolds. We prove that for a toroidal degeneration of canonical polarized manifolds with the total space $\mathbb{Q}$-factorial, the K\"{a}hler-Einstein metrics on fibers collapse to a lower dimensional complete Riemannian manifold in the pointed Gromov-Hausdorff sense by suitably choosing the base points. Furthermore, the most collapsed limit is a real affine K\"{a}hler manifold.