An Analytic Proof of the Stable Reduction Theorem

TL;DR

This paper provides a new analytic proof of the stable reduction theorem for complex curves of genus at least 2, utilizing Kähler-Einstein metrics to understand the limiting stable curves at punctures.

Contribution

It introduces an analytic approach using Kähler-Einstein metrics to prove the stable reduction theorem, offering an alternative to algebraic methods.

Findings

Established a new proof of the stable reduction theorem for complex curves.

Demonstrated the use of Kähler-Einstein metrics in understanding degenerations.

Provided insights into the geometric structure of degenerating families of curves.

Abstract

The stable reduction theorem says that a family of curves of genus $g\geq 2$ over a punctured curve can be uniquely completed (after possible base change) by inserting certain stable curves at the punctures. We give a new proof of this result for curves defined over $\mathbb{C}$ using the K\"ahler-Einstein metrics on the fibers to obtain the limiting stable curves at the punctures.