Gauge-Fixing Constant Scalar Curvature Equations on Ruled Manifolds and the Futaki Invariants

Hong, Ying Ji

doi:10.4310/jdg/1090351123

Cited by 18 publications

(36 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

See 1 more Smart Citation

Construction of constant scalar curvature Kähler cone metrics

Keller

Zheng

2018

Proc. London Math. Soc.

View full text Add to dashboard 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.

show abstract

“…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%

“…In view of it is natural to ask whether Theorem admits a generalization in the case the base manifold

B

admits non‐trivial holomorphic vector fields. As in the smooth case, this will not be automatically happen, and an extra condition on

(B, D)

will be required.…”

Section: Further Applications and Remarksmentioning

confidence: 99%

Construction of constant scalar curvature Kähler cone metrics

Keller

Zheng

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…In particular we are not assuming ω is a unique solution of the twisted extremal equation. We shall see in Remark 11 that the uniqueness assumption in Fine's result is analogous to Hong's assumption that the base has no automorphisms in Hong's first result [18], and thus the corollary above is analogous to Hong's second result [19]. Surprisingly, while Hong had to assume that the vector bundle E he considers is Aut(X, L)-invariant in [19], we do not need to make any such an assumption on the fibration X → B.…”

Section: Introductionmentioning

confidence: 89%

“…We shall see in Remark 11 that the uniqueness assumption in Fine's result is analogous to Hong's assumption that the base has no automorphisms in Hong's first result [18], and thus the corollary above is analogous to Hong's second result [19]. Surprisingly, while Hong had to assume that the vector bundle E he considers is Aut(X, L)-invariant in [19], we do not need to make any such an assumption on the fibration X → B. Thus our main result proves also the analogous result of Hong's most recent work described above [20], and also that of Lu-Seyyedali [26].…”

Section: Introductionmentioning

confidence: 89%

“…Then Hong proves that P(E) admits a cscK metric in the class rL + O P(E) (1) for r 0 [18], where we have used additive notation for the tensor product of line bundles, and suppressed pullbacks. In later work Hong relaxes the assumption on the automorphisms, showing that provided E is Aut(X, L)-invariant then the class rL + O P(E) (1) still admits a cscK metric provided the so-called Futaki invariant vanishes on P(E) [19]. Most recently Hong has proved that the analogous result holds even if E is not Aut(X, L)-invariant, however in this case E is required to satisfy a certain finite-dimensional stability condition [20].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Extremal metrics on fibrations

Dervan

Sektnan

2019

Proc. London Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

Some Remarks About Chow, Hilbert and K-stability of Ruled Threefolds

Keller

2015

Trends in Mathematics

View full text Add to dashboard Cite

Gauge-Fixing Constant Scalar Curvature Equations on Ruled Manifolds and the Futaki Invariants

Cited by 18 publications

References 0 publications

Construction of constant scalar curvature Kähler cone metrics

Construction of constant scalar curvature Kähler cone metrics

Extremal metrics on fibrations

Some Remarks About Chow, Hilbert and K-stability of Ruled Threefolds

Contact Info

Product

Resources

About