Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2\pi\ and completion of the main proof

@article{Chen2013KahlerEinsteinMO,
  title={Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2\pi\ and completion of the main proof},
  author={Xiuxiong Chen and Simon K. Donaldson and Song Sun},
  journal={arXiv: Differential Geometry},
  year={2013}
}
This is the third and final paper in a series which establish results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2\pi. We also put all our technical results together to complete the proof of the main theorem that if a K-stable Fano manifold admits a Kahler-Einstein metric. 
View PDF on arXiv
318 Citations
Highly Influential Citations
6
Background Citations
27
Methods Citations
16
Results Citations
5

318 Citations

Regularity of Kähler–Ricci flows on Fano manifolds

In this paper, we will establish a regularity theory for the Kähler–Ricci flow on Fano n-manifolds with Ricci curvature bounded in Lp-norm for some $${p > n}$$p>n. Using this regularity theory, we…

Kahler-Einstein metrics on symmetric Fano T-varieties

Kähler–Einstein metrics along the smooth continuity method

We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler–Einstein metric. This is a strengthening…

Kähler–Einstein metrics along the smooth continuity method

We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler–Einstein metric. This is a strengthening…

Conic Kähler–Einstein metrics along simple normal crossing divisors on Fano manifolds

Schwarz lemma for conical Kähler metrics with general cone angles

The Schwarz--Pick lemma is a fundamental result in complex analysis. It is well-known that Yau generalized it to the higher dimensional manifolds by applying his maximum principle for complete…

On the Curvature of Conic Kähler–Einstein Metrics

We prove a regularity result for Monge–Ampère equations degenerate along smooth divisor on Kähler manifolds in Donaldson’s spaces of $$\beta $$β-weighted functions. We apply this result to study the…

Stability of the conical Kähler-Ricci flows on Fano manifolds

Abstract In this paper, we study stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle along the divisor, then for…

Higher regularity for singular K\"ahler-Einstein metrics

We study singular Kähler-Einstein metrics that are obtained as non-collapsed limits of polarized Kähler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally…

On the moduli of Kähler-Einstein Fano manifolds

We prove that Kahler-Einstein Fano manifolds with finite automorphism groups form Hausdorff moduli algebraic space with only quotient singularities. We also discuss the limits as Q-Fano varieties…
...

References

SHOWING 1-10 OF 50 REFERENCES

Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2

This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities…

Kahler-Einstein metrics with edge singularities

This article considers the existence and regularity of Kahler-Einstein metrics on a compact Kahler manifold $M$ with edge singularities with cone angle $2\pi\beta$ along a smooth divisor $D$. We…

Kähler-Einstein metrics with positive scalar curvature

Abstract. In this paper, we prove that the existence of Kähler-Einstein metrics implies the stability of the underlying Kähler manifold in a suitable sense. In particular, this disproves a…

Conical Kähler–Einstein Metrics Revisited

In this paper we introduce the “interpolation–degeneration” strategy to study Kähler–Einstein metrics on a smooth Fano manifold with cone singularities along a smooth divisor that is proportional to…

Kähler Metrics with Cone Singularities Along a Divisor

We develop some analytical foundations for the study of Kahler metrics with cone singularities in codimension one. The main result is an analogue of the Schauder theory in this setting. In the later…

Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities

This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.

Extremal Kähler Metrics II

Given a compact, complex manifold M with a Kahler metric, we fix the deRham cohomology class Ω of the Kahler metric, and consider the function space ℊΩ of all Kahler metrics in M in that class. To…

On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*

Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the…

SINGULAR K ¨ AHLER-EINSTEIN METRICS

We study degenerate complex Monge-Ampere equations of the form (! +dd c ') n = e t' µ where ! is a big semi-Kahler form on a compact Kahler manifold X of dimension n, t 2 R + , and µ = f! n is a…

Connections withLP bounds on curvature

We show by means of the implicit function theorem that Coulomb gauges exist for fields over a ball inRn when the integralLn/2 field norm is sufficiently small. We then are able to prove a weak…