Compact Moduli Spaces of Del Pezzo Surfaces and K\"ahler-Einstein metrics

TL;DR

This paper establishes a correspondence between the metric and algebraic compactifications of moduli spaces of Kähler-Einstein Del Pezzo surfaces, providing a classification of degenerations and confirming Tian's theorem.

Contribution

It proves the equivalence of Gromov-Hausdorff and algebro-geometric compactifications for these moduli spaces, combining algebraic and differential geometry methods.

Findings

Gromov-Hausdorff compactification matches algebraic compactification

Classifies degenerations of Kähler-Einstein Del Pezzo surfaces

Confirms Tian's theorem on existence of Kähler-Einstein metrics

Abstract

We prove that the Gromov-Hausdorff compactification of the moduli space of Kahler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the existence of Kahler-Einstein metrics on smooth Del Pezzo surfaces and classifies the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.