On regularization of plurisubharmonic functions on manifolds

Błocki, Zbigniew; Kołodziej, Sławomir

doi:10.1090/s0002-9939-07-08858-2

Cited by 148 publications

(180 citation statements)

References 10 publications

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“…the points in the set D in the Gromov-Hausdorff limit. The arguments in [42,Proposition 11,12,13] An additional important fact used several times is that by Cheeger-ColdingTian [15], no tangent cone of the form C γ ×C n−1 can form in the Gromov-Hausdorff limit of a sequence of Kähler metrics with bounded Ricci curvature. The analogous result with the bound on Ricci curvature replaced by a bound on Ric(ω) − L v ω was shown by Tian-Zhang [47], and it also follows from the more recent work of Cheeger-Naber [16] in the general Riemannian case.…”

Section: Proposition 21mentioning

confidence: 99%

“…First, if we let η = ω ′ + √ −1ds ∧ ds, then by the approximation theorem of Demailly [23] (see also Blocki-Kolodziej [13]) there exists a decreasing sequence of smooth metrics ρ s,ǫ ց p * τ s such that √ −1∂∂ s,W ′ (ρ s,ǫ ) ≥ −Cη. By averaging we can also suppose that ρ s,ǫ are independent of Re(s) and T -invariant.…”

Section: Reductivity Of the Automorphism Group And Vanishing Of The Fmentioning

confidence: 99%

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Kähler–Einstein metrics along the smooth continuity method

2016

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Abstract. We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-DonaldsonSun [17], and can be used to obtain new examples of Kähler-Einstein manifolds. We also give analogous results for twisted Kähler-Einstein metrics and Kahler-Ricci solitons.

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Section: Proposition 21mentioning

confidence: 99%

Section: Reductivity Of the Automorphism Group And Vanishing Of The Fmentioning

confidence: 99%

Kähler–Einstein metrics along the smooth continuity method

2016

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“…• Approximate ϕ 0 by a decreasing sequence (ϕ 0,j ) of smooth and strictly ω-psh functions by using the regularization result of Demailly [Dem92,BK07]. There exists unique solutions ϕ t,j ∈ P SH(X, ω) ∩ C ∞ (X) to the flow above with initial data ϕ 0,j .…”

Section: ) and ϕ T Is Uniformly Bounded And Converges Tomentioning

confidence: 99%

“…Convergence in L 1 . We approximate ϕ 0 by a decreasing sequence ϕ 0,j of smooth ω-psh fuctions (using [Dem92] or [BK07]). Denote by ϕ t,j the smooth family of θ t -psh functions satisfying on…”

Section: T D Tômentioning

confidence: 99%

Regularizing properties of complex Monge–Ampère flows

Tô

2017

Journal of Functional Analysis

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Abstract. We study the regularizing properties of complex MongeAmpère flows on a Kähler manifold (X, ω) when the initial data are ω-psh functions with zero Lelong number at all points. We prove that the general Monge-Ampère flow has a solution which is immediately smooth. We also prove the uniqueness and stability of solution. IntroductionLet (X, ω) be a compact Kähler manifold of complex dimension n and α ∈ H 1,1 (X, R) a Kähler class with ω ∈ α. Let Ω be a smooth volume form on X. Denote by (θ t ) t∈[0,T ] a family of Kähler forms on X, and assume that θ 0 = ω. The goal of this note is to prove the regularizing and stability properties of solutions to the following complex Monge-Ampère flowwhere F is a smooth function and ϕ(0, z) = ϕ 0 (z) is a ω-plurisubharmonic (ω-psh) function with zero Lelong numbers at all points.One motivation for studying this Monge-Ampère flow is that the Käler-Ricci flow can be reduced to a particular case of (CM AF ). When F = F (z) and θ t = ω + tχ, where χ = η − Ric(ω), then (CM AF ) is the local potential equation of the twisted Kähler-Ricci flow

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“…Other extensions of the Bedford-Taylor theory to the manifold setting can be found in [65] and [14].…”

Section: ∂∂φ)mentioning

confidence: 99%

Lectures on Stability and Constant Scalar Curvature

Phong¹,

Sturm²

2007

Current Developments in Mathematics

108

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An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kähler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite-dimensional GIT, and conditions on the closure of orbits of almost-complex structures under the diffeomorphism group. Related analytic methods are also discussed, including estimates for energy functionals, Tian-Yau-Zelditch approximations, estimates for moment maps, complex Monge-Ampère equations and pluripotential theory, and the Kähler-Ricci flow.

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On regularization of plurisubharmonic functions on manifolds

Abstract: Abstract. We study the question of when a γ-plurisubharmonic function on a complex manifold, where γ is a fixed (1, 1)-form, can be approximated by a decreasing sequence of smooth γ-plurisubharmonic functions. We show in particular that it is always possible in the compact Kähler case.

Cited by 148 publications

References 10 publications

Kähler–Einstein metrics along the smooth continuity method

Kähler–Einstein metrics along the smooth continuity method

Regularizing properties of complex Monge–Ampère flows

Lectures on Stability and Constant Scalar Curvature

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