Extremal metrics on blowups

Arezzo, Claudio; Pacard, Frank; Singer, Michael F.

doi:10.1215/00127094-2011-001

Cited by 59 publications

(144 citation statements)

References 30 publications

(62 reference statements)

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“…This is roughly a Kähler analogue of the strategy of Stoppa-Székelyhidi [46], which in turn is related to Stoppa's strategy in the cscK case [44]. The key analytic result will be the following, due to Arezzo-Pacard-Singer [3]. We use the formulation of [46,Theorem 8].…”

Section: Theorem 46 If (X [ω]) Admits An Extremal Metric Then It Imentioning

confidence: 99%

Relative K-stability for Kähler manifolds

Dervan

2017

Math. Ann.

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We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kähler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. Along the way, we prove a general L p lower bound on the Calabi functional involving test configurations and their associated numerical invariants, answering a question of Donaldson. When the Kähler manifold is projective, our definition of relative K-stability is stronger than the usual definition given by Székelyhidi. In particular our result strengthens the known results in the projective case (even for constant scalar curvature Kähler metrics), and rules out a well known counterexample to the "naïve" version of the Yau-Tian-Donaldson conjecture in this setting.

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Section: Theorem 46 If (X [ω]) Admits An Extremal Metric Then It Imentioning

confidence: 99%

Relative K-stability for Kähler manifolds

Dervan

2017

Math. Ann.

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“…We continue our study [32] of extremal metrics on blown-up manifolds, following the work of Arezzo and Pacard [1,2] and Arezzo et al [3]. See Pacard [21] for a survey and see also LeBrun and Singer [18], Rollin and Singer [23], Tipler [35] and Biquard and Rollin [4] for related work.…”

Section: Introductionmentioning

confidence: 95%

Blowing up extremal Kähler manifolds II

Székelyhidi

2014

Invent. math.

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This is a continuation of the work of Arezzo-Pacard-Singer and the author on blowups of extremal Kähler manifolds. We prove the conjecture stated in Székelyhidi (Duke Math J 161 (8):1411-1453, 2012), and we relate this result to the K-stability of blown up manifolds. As an application we prove that if a Kähler manifold M of dimension >2 admits a constant scalar curvature (cscK) metric, then the blowup of M at a point admits a cscK metric if and only if it is K-stable, as long as the exceptional divisor is sufficiently small.

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“…Theorem 2.4 (Arezzo-Pacard-Singer [3]). Suppose that M admits an extremal metric in c 1 (L), and let T be a maximal torus of automorphisms of (M, L).…”

Section: Relative K-polystabilitymentioning

confidence: 99%

“…Indeed we can choose an extremal metric ω on M such that the isometry group of ω contains a compact maximal torus T R , which is contained in the complex torus T . In the notation of [3] we let K = T R , and let k be its Lie algebra. Since K is a maximal torus, any K-invariant holomorphic hamiltonian vector field lies in k. Moreover if we write S(ω) for the scalar curvature then by Calabi's theorem [5] the vector field J ∇S(ω) lies in the centre of the Lie algebra of Killing fields, so it also lies in k. This allows us to apply [3, Theorem 2.1], and we get the stated result.…”

Section: Relative K-polystabilitymentioning

confidence: 99%