Constant Hermitian scalar curvature equations on ruled manifolds

Hong, Ying-Ji

doi:10.4310/jdg/1214425636

Cited by 41 publications

(46 citation statements)

References 17 publications

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“…Our next main result is a construction theorem of cscK metrics with cone singularities in Kähler classes (that may not be integral) over projective bundles, which generalizes the main result of . It is also an application of Theorems and but requires much more work.…”

Section: Introductionmentioning

confidence: 70%

“…Let us remember that we know from Hong's techniques see [, Section II; , Theorem 3.1]. Lemma On the regular part of

P E^{*}

, we know the scalar curvature of the metric

ω_{k}

\begin{matrix} true \\ right S (ω_{k}) (false[v false]) = & left r (r - 1) + \frac{1}{k} (() π^{*} S false(ω_{B} false) + 2 r \frac{\sqrt{- 1}}{2 π} {normalΛ}_{ω_{B}} tr ({false[F_{h_{E}} false]}^{0} \frac{v \otimes v^{*_{h_{E}}}}{{∥ v ∥}^{2}})) \\ left + O ((), \frac{1}{k^{2}}), \end{matrix}

where

r = rk (E)

false[v false] \in P E^{*}

and

{false[. false]}^{0}

denotes the trace‐free part.…”

Section: Construction Of Csck Cone Metrics Over Projective Bundlesmentioning

confidence: 99%

“…In view of Theorem , we will deform the metric

ω_{k}

by deforming the metrics

ω_{B}

and

h_{E}

in order to obtain, after another deformation, a sequence of Kähler cone metrics that have almost constant scalar curvature. Then we will apply the contraction mapping theorem, following the main idea of . Proposition Assume

ω_{B}

is cscK with conical singularities along

D

with Hölder exponent

α

and angle

2 π β

satisfying Condition and such that

L i e (A u t_{D} false(B, [ω_{B}] false))

is trivial.…”

Section: Construction Of Csck Cone Metrics Over Projective Bundlesmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 70%

“…Let us remember that we know from Hong's techniques see [, Section II; , Theorem 3.1]. Lemma On the regular part of

P E^{*}

, we know the scalar curvature of the metric

ω_{k}

\begin{matrix} true \\ right S (ω_{k}) (false[v false]) = & left r (r - 1) + \frac{1}{k} (() π^{*} S false(ω_{B} false) + 2 r \frac{\sqrt{- 1}}{2 π} {normalΛ}_{ω_{B}} tr ({false[F_{h_{E}} false]}^{0} \frac{v \otimes v^{*_{h_{E}}}}{{∥ v ∥}^{2}})) \\ left + O ((), \frac{1}{k^{2}}), \end{matrix}

where

r = rk (E)

false[v false] \in P E^{*}

and

{false[. false]}^{0}

denotes the trace‐free part.…”

Section: Construction Of Csck Cone Metrics Over Projective Bundlesmentioning

confidence: 99%

“…In view of Theorem , we will deform the metric

ω_{k}

by deforming the metrics

ω_{B}

and

h_{E}

ω_{B}

is cscK with conical singularities along

D

with Hölder exponent

α

and angle

2 π β

satisfying Condition and such that

L i e (A u t_{D} false(B, [ω_{B}] false))

is trivial.…”

Section: Construction Of Csck Cone Metrics Over Projective Bundlesmentioning

confidence: 99%

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Construction of constant scalar curvature Kähler cone metrics

Keller

Zheng

2018

Proc. London Math. Soc.

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“…The resulting time-dependent Hermitian Yang-Mills flow (which we sometimes simply refer to as Hermitian Yang-Mills flow or abbreviate as HYMF) arises naturally in the construction of adiabatic approximations to Calabi flow on ruled manifolds (this can be thought of a parabolic versions of Hong's construction of adiabatic cscK metrics on ruled manifolds in [7,8]). In brief, if π : E → X is a holomorphic line bundle over a compact Kähler base and ω r = ω 0 (h) + r π * ω X is an adiabatic family of metrics on the ruled manifold PE specified by a Kähler metric ω X on X and the curvature ω 0 (h) of the relative hyperplane bundle defined by a Hermitian metric h on E, then a first order (in r −1 ) approximation to Calabi flow of the adiabatic metric ω r is given by evolving ω X by Calabi flow on the base and h by HYMF with respect to the now time-dependent base metric ω X (t).…”

Section: Introductionmentioning

confidence: 99%

Time-dependent Hermitian Yang–Mills flow on surfaces

Pook

2015

Geom Dedicata

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We investigate a generalisation of Hermitian Yang-Mills flow in which the base metric itself is allowed to depend on the time-parameter. For technical reasons we restrict our attention to the case of a holomorphic vector bundles E over compact Riemann surfaces which is assumed to be slope stable with respect to a fixed Kähler class τ on the base. We show that if ω(t) is a family of Kähler metrics in τ converging to a limit metric ω ∞ at an exponential rate, then starting at a smooth initial Hermitian metric h 0 on E, the Hermitian Yang-Mills flow with respect to the time-dependent metric ω(t) admits a unique long-time solution h(t) converging exponentially to a ω ∞ -Hermite-Einstein metric. The proof utilises an extension of Donadson's techniques used to treat the case where the base metrics does not depend on the time-parameter.

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“…To motivate our work, we first review some older constructions of extremal Kähler metrics. The first general construction of such metrics is due to Hong , who considered fibrations

π : P (E) \to B

, where

P (E)

denotes the projectivisation of a vector bundle

E

on a compact complex manifold

B

. Suppose

B

admits a line bundle

L

with a cscK metric, and suppose in addition that

E

admits a Hermite–Einstein metric.…”

Section: Introductionmentioning

confidence: 99%

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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Constant Hermitian scalar curvature equations on ruled manifolds

Cited by 41 publications

References 17 publications

Construction of constant scalar curvature Kähler cone metrics

Construction of constant scalar curvature Kähler cone metrics

Time-dependent Hermitian Yang–Mills flow on surfaces

Extremal metrics on fibrations

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