$L^1$ metric geometry of big cohomology classes

Darvas, Tamás; Nezza, Eleonora Di; Lu, Chinh H.

doi:10.5802/aif.3236

Cited by 37 publications

(25 citation statements)

References 32 publications

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“…Several proofs more or less available in the literature will be included in this preliminary section for the convenience of the reader. For more details we refer to [3,5,[8][9][10][11][12].…”

Section: Preliminariesmentioning

confidence: 99%

Comparison of Monge–Ampère capacities

2021

Ann. Polon. Math.

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Section: Preliminariesmentioning

confidence: 99%

Comparison of Monge–Ampère capacities

2021

Ann. Polon. Math.

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“…We first recall a few known facts on pluripotential theory in big cohomology classes. We refer the reader to [2,8,18,19,20,21,22] for more details.…”

Section: The Case Of Big Cohomology Classes On Kähler Manifoldsmentioning

confidence: 99%

Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds

Lu¹,

Phung²,

Tô³

2022

Annales De l'Institut Fourier

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Let (X, ω) be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge-Ampère equations with right-hand side in L p , p > 1. Using this we prove that the solutions are Hölder continuous with the same exponent as in the Kähler case by Demailly-Dinew-Guedj-Kołodziej-Pham-Zeriahi. Our techniques also apply to the setting of big cohomology classes on compact Kähler manifolds.Résumé. -Soit (X, ω) une variété Hermitienne compacte de dimension n. On établit la stabilité des solutions des équations de Monge-Ampère avec second membre dans L p , p > 1. En utilisant ce résultat on montre que les solutions sont continues höldériennes avec le même exposant que celui obtenu dans le cas Kählérien par Demailly-Dinew-Guedj-Kołodziej-Pham-Zeriahi. Notre méthode s'applique également dans le contexte des classes de cohomologie sur une variété Kählérienne.

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“…Proof. If u, v have ψ-relative minimal singularities then it is the content of Theorem 4.10 in [DDNL17b], while in the general case the proof is the same to that of Proposition 2.2 in [DDNL18a] replacing V θ with ψ, using the Comparison Principle of Proposition 2.1 and the fact that for any w ∈ E 1 (X, ω, ψ)…”

Section: Moreover We Also Have the Following Propertiesmentioning

confidence: 99%

$L^{1}$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds

Trusiani

2019

Preprint

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Given (X, ω) compact Kähler manifold and ψ ∈ M + ⊂ P SH(X, ω) a model type envelope with non-zero mass, i.e. a fixed potential determing some singularities such that X (ω +dd c ψ) n > 0, we prove that the ψ−relative finite energy class E 1 (X, ω, ψ) becomes a complete metric space if endowed of a distance d which generalizes the well-known d 1 distance on the space of Kähler potentials. Later, for A ⊂ M + total ordered, we equip the set X A := ψ∈A E 1 (X, ω, ψ) of a natural distance d A which coincides with the distance d on E 1 (X, ω, ψ) for any ψ ∈ A. We show that X A , d A is a complete metric space. As a consequence, assuming ψ k ց ψ and ψ k , ψ ∈ M + , we also prove that E 1 (X, ω, ψ k ), d converges in a Gromov-Hausdorff sense to E 1 (X, ω, ψ), d and that it is possible to define a direct system E 1 (X, ω, ψ k ), P k,j in the category of metric spaces whose direct limit is dense into E 1 (X, ω, ψ), d .

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$L^1$ metric geometry of big cohomology classes

Cited by 37 publications

References 32 publications

Comparison of Monge–Ampère capacities

Comparison of Monge–Ampère capacities

Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds

$L^{1}$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds

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