Compact Moduli Spaces of Del Pezzo Surfaces and Kähler-Einstein metrics

2012/10/02 by Yuji Odaka, Odaka, Yuji, Cristiano Spotti +3 · 8 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1210.0858

openalex publication_date 2012/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Degenerations of del Pezzo surfaces I
• Special test configurations and K-stability of Fano varieties

Cited by

• Yang-Mills energy quantization over non-collapsed degenerating Einstein manifolds and applications
• Optimal bounds for local volumes of threefold singularities
• K-Moduli of Fano Threefolds of Family 3.3
• On multiscale aspects of Kähler-Einstein metrics and algebraic geometry
• Compact moduli space of Kahler-Einstein Fano varieties
• K-stability of del Pezzo surfaces with a single quotient singularity
• Weyl Curvature, Del Pezzo Surfaces, and Almost-Kaehler Geometry
• Completion of the moduli space for polarized Calabi-Yau manifolds

Related

• Existence of Kähler-Einstein metrics and multiplier ideal sheaves on del Pezzo surfaces
• Degenerations of Kähler-Einstein Fano Manifolds
• Explicit Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds
• Compact moduli space of Kahler-Einstein Fano varieties
• Canonical Metrics on the Moduli Space of Riemann Surfaces II