Fibrations with constant scalar curvature Kähler metrics and the CM-line bundle

Fine, Joël

doi:10.4310/mrl.2007.v14.n2.a7

Cited by 33 publications

(40 citation statements)

References 19 publications

(26 reference statements)

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“…As the Kähler classes degenerate to a Kähler class on the base Y , the cscK metrics in Theorem 1.2 should converge to a twisted constant scalar curvature metric on Y as shown in [7,8] in the case when the fibration map Φ is submersion.…”

Section: Introductionmentioning

confidence: 96%

A remark on constant scalar curvature Kähler metrics on minimal models

Jian

Shi

Song

2019

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 96%

A remark on constant scalar curvature Kähler metrics on minimal models

Jian

Shi

Song

2019

Proc. Amer. Math. Soc.

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“…As in Fine's work [14,15], applying this to the horizontal component of ρ gives a (1,1)-form α on B, which is closed. Setting ρ H to be the horizontal component of ρ, explicitly we have…”

Section: Fibrationsmentioning

confidence: 83%

“…When the fibres X b are curves, Lemma 3.5 was noted by Fine [14,Theorem 3.5]. In general Fine remarks in [15] that [α] = [ω W P ], what we wish to point out here is that even the forms themselves are equal.…”

Section: The Weil-petersson Metricmentioning

confidence: 84%

“…One can generalise this as follows. For a

(p, p)

‐form

η

X

one obtains a

(p - m, p - m)

form

\int_{X / B} η

B

by first using the submersion structure to write

η = ψ \land π^{*} κ

for

κ

(p - m, p - m)

‐form on

B

, and then defining

\int_{X / B} η = (\int_{X / B} ψ) κ .

As in Fine's work , applying this to the horizontal component of

ρ

gives a (1,1)‐form

α

B

, which is closed. Setting

ρ_{H}

to be the horizontal component of

ρ

, explicitly we have

α = - \frac{\int_{X / B} false(ρ_{H} \land ω_{0}^{m} false)}{\int_{X / B} ω_{0}^{m}} .

…”

Section: Preliminariesmentioning

confidence: 98%

“…Proof This is essentially proved in [, Lemma 2.3]; we recall Fine's argument for the reader's convenience. Recall that

α

is defined by first taking the horizontal component

ρ_{H}

ρ

and defining

α = - \frac{\int_{X / B} false(ρ_{H} \land ω^{m} false)}{\int_{X / B} ω^{m}} .

On each fibre, the vertical component

ρ_{V}

ρ

is the Ricci curvature of

ω_{b}

[, Lemma 3.3].…”

Section: Lifting Vector Fieldsmentioning

confidence: 99%

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Extremal metrics on fibrations

Dervan

Sektnan

2019

Proc. London Math. Soc.

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Consider a fibred compact Kähler manifold X endowed with a relatively ample line bundle, such that each fibre admits a constant scalar curvature Kähler (cscK) metric and has discrete automorphism group. Assuming the base of the fibration admits a twisted extremal metric where the twisting form is a certain Weil–Petersson type metric, we prove that X admits an extremal metric for polarisations making the fibres small. Thus, X admits a cscK metric if and only if the Futaki invariant vanishes. This extends a result of Fine who proved this result when the base admits no continuous automorphisms. As consequences of our techniques, we obtain analogues for maps of various fundamental results for varieties: if a map admits a twisted cscK metric, then its automorphism group is reductive; a twisted extremal metric is invariant under a maximal compact subgroup of the automorphism group of the map; there is a geometric interpretation for uniqueness of twisted extremal metrics on maps.

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Optimal Symplectic Connections on Holomorphic Submersions

Dervan¹,

Sektnan

2020

Comm Pure Appl Math

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The main result of this paper gives a new construction of extremal Kähler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of Hong, Fine and others. The principal new ingredient is a novel geometric partial differential equation on such fibrations, which we call the optimal symplectic connection equation. We begin with a smooth fibration for which all fibres admit a constant scalar curvature Kähler metric. When the fibres admit automorphisms, such metrics are not unique in general, but rather are unique up to the action of the automorphism group of each fibre. We define an equation which, at least conjecturally, determines a canonical choice of constant scalar curvature Kähler metric on each fibre. When the fibration is a projective bundle, this equation specialises to asking that the hermitian metric determining the fibrewise Fubini‐Study metric is Hermite‐Einstein. Assuming the existence of an optimal symplectic connection and the existence of an appropriate twisted extremal metric on the base of the fibration, we show that the total space of the fibration itself admits an extremal metric for certain polarisations making the fibres small. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

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Fibrations with constant scalar curvature Kähler metrics and the CM-line bundle

Cited by 33 publications

References 19 publications

A remark on constant scalar curvature Kähler metrics on minimal models

A remark on constant scalar curvature Kähler metrics on minimal models

Extremal metrics on fibrations

Optimal Symplectic Connections on Holomorphic Submersions

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