Uniqueness of constant scalar curvature Kähler metrics with cone singularities. I: reductivity

Proc. London Math. Soc.

2018

Self Cite

Abstract. Over a compact Kähler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kähler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kähler metrics with conic singularities: existence result under small deformations of Kähler classes, existence result over a Fano manifold, existence result over certain ruled manifolds. In this last case, we consider the projectivisation of a parabolic stable holomorphic bundle. This leads us to prove that the existing Hermitian-Einstein metric on this bundle enjoys a regularity property along the divisor on the base.

“…But we can see that the higher order spaces are more complicated, since the geometry of the background metric is a priori involved. The Hölder space

C^{3, α, β}

and

C^{4, α, β}

Section: Csck Metrics With Cone Singularitiesmentioning

confidence: 99%

α

and

β

satisfy Condition .…”

Section: Csck Metrics With Cone Singularitiesmentioning

confidence: 99%

“…In fact, we have a more general property. Details can also be found in [, Section 5.3]. Proposition Assume that

ω

is a Kähler cone metric and its connection satisfies .…”

Section: Csck Metrics With Cone Singularitiesmentioning

confidence: 99%

“…The proofs of Propositions and are carried out in . Moreover both spaces

C^{3, α, β} false(X; ω false)

and

C^{4, α, β} false(X; ω false)

are Banach spaces, as proved in [, Section 5]. Remark It is natural to use the model metric

ω_{D}

in both Propositions and , since it satisfies all required conditions.…”

Section: Csck Metrics With Cone Singularitiesmentioning

confidence: 99%

“…We will need a certain restriction on the cone angle

2 π β

and the Hölder exponent

α

, namely that

0 < β < \frac{1}{2}; α β < 1 - 2 β .

This restriction appeared in previous works, for example, in and is required to have regularity results.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Construction of constant scalar curvature Kähler cone metrics

Keller

Proc. London Math. Soc.

2018

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Geodesics in the Space of Kähler Cone Metrics II: Uniqueness of Constant Scalar Curvature Kähler Cone Metrics

2019

Comm Pure Appl Math

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In this article, we give a complete construction of geodesics in the space of Kähler cone metrics (cone geodesics), and we address the problem on the uniqueness of constant scalar curvature Kähler (cscK) cone metrics when the cone angle β stays in the whole interval (0, 1]. The part β∈[)12,1 requires new weighted function spaces and new analytic techniques. We determine the asymptotic behavior of both cone geodesics and cscK cone metrics, prove the reductivity of the automorphism group, and establish the linear theory for the Lichnerowicz operator, which immediately implies the openness of the path deforming the cone angles of cscK cone metrics. © 2019 Wiley Periodicals, Inc.

Generalized Matsushima’s theorem and Kähler–Einstein cone metrics

2018

Calc. Var.

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In this paper, we prove Matsushima's theorem for Kähler-Einstein metrics on a Fano manifold with cone singularities along a smooth divisor that is not necessarily proportional to the anti-canonical class. We then give an alternative proof of uniqueness of Kähler-Einstein cone metrics by the continuity method. Moreover, our method provides an existence theorem of Kähler-Einstein cone metrics with respect to conic Ding functional.Remark 1.2. For the klt -pair, Chen-Donaldson-Sun [13] proved that the automorphism group is reductive. However, they required the uniqueness of weak Kähler-Einstein metrics in their proof.Remark 1.3. In [10], Cheltsov-Rubinstein also announced a result for extremal cone metrics, but their method is based on an expansion formula for edge metrics, which is very different from ours.Based on this reductivity result, we can extend Bando-Mabuchi's celebrated work [1] to conic setting and prove the uniqueness of Kähler-Einstein cone metris by applying the continuity path, which connects the Kähler-Einstein cone metric ω ϕ to a given Kähler cone metric ω. I.e. for any t ∈ [0, 1],And we proved the following Theorem 1.4. The Kähler-Einstein cone metric is unique up to automorphisms.The way to prove uniqueness is first to establish a continuity path connecting a general Kähler cone metric to our target, i.e. a Kähler-Einstein cone metric. The difficulties are to prove openness and closedness along the path in Donaldson's C 2,α β space: here openness on [0, 1) follows from a Bochner type formula with contradiction argument. Thus we are able to carry on the implicit function theorem on [0, 1) and the apriori estimates on [0, τ ] for a small fixed τ > 0.Meanwhile, in order to prove closedness on [τ, 1], everything is boiled down to prove the zero order estimate (Section 4.3.1) and the higher order estimates (Section 4.3.2). We first show that the zero order estimate of the continuity path on [τ, 1] requires only the uniform bound