Compact moduli spaces of Del Pezzo surfaces and Kähler–Einstein metrics

Abstract

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