Degenerations of K\"ahler-Einstein Fano Manifolds

TL;DR

This thesis explores how Fano manifolds with Kähler-Einstein metrics degenerate in the Gromov-Hausdorff sense and examines the implications for algebraic geometry, especially regarding moduli space compactifications.

Contribution

It provides new insights into the degeneration behavior of Kähler-Einstein Fano manifolds and links metric degenerations with algebraic geometric structures.

Findings

Characterization of degenerations in Gromov-Hausdorff topology

Connections between metric degenerations and algebraic moduli spaces

New results on compactification of moduli spaces of Fano manifolds

Abstract

In this Thesis, I investigate how Fano manifolds equipped with a Kahler-Einstein metric can degenerate as metric spaces (in the Gromov-Hausdorff topology) and some of the relations of this question with Algebraic Geometry, in particular in the direction of the study of moduli spaces and their compactifications.