Constant scalar curvature Kähler metrics on fibred complex surfaces

Fine, Joël

doi:10.4310/jdg/1115669591

Cited by 78 publications

(195 citation statements)

References 11 publications

(16 reference statements)

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“…Indeed, for the blow-up construction a m−1 j ε 2m−2 gives the volume of the exceptional divisor E j (up to a universal constant depending only on the dimension). The role of ε in our results has a direct analogue in Fine-Hong's papers [19], [24], [25], blowing up and desingularizing kähler manifolds 187 replacing the exceptional divisor with the fiber of the projection onto the Riemann surface or the projectivized fiber of the vector bundle.…”

Section: Introductionmentioning

confidence: 93%

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Blowing up and desingularizing constant scalar curvature Kähler manifolds

2006

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Section: Introductionmentioning

confidence: 93%

“…Let ∆ 0 denote the Laplacian in C m with its standard Kähler form. Using (3), it is easy to check that, near each p j , (19) holds for some function v if and only if…”

Section: Analysis Of the Operators Defined On (M * ω)mentioning

confidence: 99%

Blowing up and desingularizing constant scalar curvature Kähler manifolds

2006

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“…Indeed, we wish to find for

k > > 0

a potential

θ \in C^{4, α, β} false(ω_{D} false)

such that

{\overset{ω}{true}}_{k, p} + \sqrt{- 1} \partial \bar{\partial} θ > 0

and we have over

X

S ({\overset{ω}{true}}_{k, p} + \sqrt{- 1} \partial \bar{\partial} θ) = C s t .

The linearization operator

scriptL

of the previous equation is given by

θ \mapsto {L ic}_{{\tilde{ω}}_{k, p}} θ + \nabla_{{\tilde{ω}}_{k, p}} θ . \nabla_{{\tilde{ω}}_{k, p}} S false({\tilde{ω}}_{k, p} false) = {L ic}_{{\tilde{ω}}_{k, p}} θ + \nabla_{{\tilde{ω}}_{k, p}} θ . O (\frac{1}{k^{p + 1}}) .

Thanks to Corollary , the kernel of the Lichnerowicz operator

{double-struckL ic}_{{\overset{ω}{true}}_{k, p}}

is trivial. Moreover, the techniques of [, Section 6.2; , Section 4.3.1] can be applied without change to get a rough lower bound on the lowest eigenvalue of this operator and this will allow us to see that

scriptL

is a Banach isomorphism. Actually, there exists a constant

C_{p} > 0

such that for any

θ \in C^{4, α, β}

…”

Section: Construction Of Csck Cone Metrics Over Projective Bundlesmentioning

confidence: 99%

Construction of constant scalar curvature Kähler cone metrics

Keller

Zheng

2018

Proc. London Math. Soc.

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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.

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“…The rest of the assumption (4.17) can be easily verified by (5.1)(cf. Lemma 2.7, 2.8 in [15]). The theorem is proved.…”

Section: Stepmentioning

confidence: 99%