# R\'eduction stable en dimension sup\'erieure [d'apr\`es Koll\'ar, Hacon-Xu...]

## TL;DR

This paper discusses recent advances in constructing moduli spaces of stable higher-dimensional algebraic varieties, extending the classical theory of stable curves, and highlights the stable reduction theorem's role in establishing their compactness.

## Contribution

It introduces new ideas for constructing moduli spaces of stable higher-dimensional varieties, including a higher-dimensional stable reduction theorem that ensures their compactness.

## Key findings

- Construction of higher-dimensional moduli spaces achieved
- Stable reduction theorem extended to higher dimensions
- Moduli spaces are shown to be compact

## Abstract

The moduli space of stable curves of Deligne and Mumford is a compactification of the moduli space of smooth curves of genus >=2 that parametrizes certain nodal curves. It is a powerful tool for the study of algebraic curves. Higher-dimensional analogues were constructed by Koll\'ar, Shepherd-Barron and Alexeev in dimension 2, and by Viehweg in the case of smooth varieties. We will explain the recent ideas allowing for the construction of these moduli spaces in general, including the stable reduction theorem in higher dimension, which reflects their compactness. L'espace de modules des courbes stables de Deligne et Mumford est une compactification de l'espace de modules des courbes lisses de genre >=2, param\'etrant certaines courbes nodales. C'est un outil puissant pour l'\'etude des courbes alg\'ebriques. Des analogues en dimension sup\'erieure ont \'et\'e construits par Koll\'ar, Shepherd-Barron et Alexeev en dimension 2, et par Viehweg dans le cas des vari\'et\'es lisses. Nous expliquerons les id\'ees r\'ecentes ayant permis la construction de ces espaces de modules en g\'en\'eral, notamment le th\'eor\`eme de r\'eduction stable en dimension sup\'erieure, qui refl\`ete leur compacit\'e.