Compact Moduli Spaces of Del Pezzo Surfaces and 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.

Final version. To appear in J. Diff. Geom

References in corpus (8)

• K-stability of constant scalar curvature polarization
• Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II
• Special test configurations and $K$-stability of Fano varieties
• K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
• Degenerations of del Pezzo surfaces I
• Ricci flow on Orbifold
• Log canonical models and variation of GIT for genus four canonical curves
• On parametrization, optimization and triviality of test configurations

Cited by in corpus (16)

• Boundedness of $\mathbb{Q}$-Fano varieties with degrees and alpha-invariants bounded from below
• On the moduli of Kahler-Einstein Fano manifolds
• Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds
• On toric geometry and K-stability of Fano varieties
• On the classification of Kahler-Ricci solitons on Gorenstein del Pezzo surfaces
• Applications of the moduli continuity method to log K-stable pairs
• Einstein Metrics, Harmonic Forms, and Symplectic Four-Manifolds
• Compact moduli space of Kahler-Einstein Fano varieties
• Moduli of cubic surfaces and their anticanonical divisors
• Fano-Ricci limit spaces and spectral convergence
• Tropical Geometric Compactification of Moduli, II - $A_g$ case and holomorphic limits -
• Bach-Flat Kaehler Surfaces
• Completion of the moduli space for polarized Calabi-Yau manifolds
• Weyl Curvature, Del Pezzo Surfaces, and Almost-Kaehler Geometry
• Separatedness of moduli of K-stable varieties
• Stability of nets of quadrics in ${\mathbb{P}}^5$ and associated discriminants