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

By Y. Odaka, Cristiano Spotti, Song Sun. 2012

162 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b842122774e66b06c136a7f3e482ec1aded9fe82

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, background): “The existence problem in the singular case is still open in general, except for the toric case [ BB13 ]; cf also [ OSS16 ] for a related problem in the case of singular Fano surfaces.”
Positivity of the CM line bundle for families of K-stable klt Fano varieties (2020, background): “Smoothable singular K-stable Del-Pezzo surfaces are classified in [93].”
On the moduli of Kähler-Einstein Fano manifolds (2012, methodology): “r weakly polarized varieties with singular cscK metrics. The connected components of M¯ cscK are all projective schemes, because the CM line bundle descends to ample line bundles on them. We refer to [OSS12] for detailed discussions on our Q-Fano varieties case which largely settled (smoothable) del”
On the K-stability of Fano varieties and anticanonical divisors (2016, background): “or the case of Fano varieties). It also turned out that such canonical Ka¨hler metrics and K-stability play crucial roles for nice moduli theory 1 (cf., [FS90, Od10, DS14, Od13]), and indeed recently [OSS16] constructed compact moduli spaces of smoothable Ka¨hler-Einstein Fano varieties of two dimen”
K-stability of Fano varieties: an algebro-geometric approach (2020, methodology): “ K-stable Fano manifolds, and all other possible limits are not K-(semi,poly)stable. This approach ﬁrst appeared explicitly in [MM93] where they solved the case of degree 4 del Pezzo surface. Then in [OSS16], all cases of smoothable surfaces were completed. In [SS17], the authors settled the case of”
Boundedness of Q-Fano varieties with degrees and alpha-invariants bounded from below (2017, background): “ty play crucial roles for construction of nice moduli spaces of certain Q-Fano varieties. For example, compact moduli spaces of smoothable Ka¨hler–Einstein Q-Fano varieties have been constructed (see [OSS16] for dimension two case and [LWX14, SSY16, Oda15] for higher dimensional case). In order to c”