An extension theorem for Kähler currents with analytic singularities

Collins, Tristan C.; Tosatti, Valentino

doi:10.5802/afst.1429

Cited by 16 publications

(18 citation statements)

References 17 publications

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“…When α is any ample class, there is a very similar theorem which has appeared in the proof of Proposition 4.13. However, the proof there relies on the difficult extension theorem in [CT14]. Here we give a simple and direct proof when X is a complex surface.…”

Section: Generalized Okounkov Bodies On Complex Surfacesmentioning

confidence: 95%

“…We choose t > 0 small enough such that ω − tY 1 is still a Kähler class. By the main theorem of [CT14], any Kähler current T ∈ (ω − tY 1 )| Y 1 with analytic singularities can be extended to a Kähler current T ∈ ω − tY 1 , thus we have…”

Section: Ya Dengmentioning

confidence: 99%

“…Here we give a simple and direct proof when X is a complex surface. Anyway, the idea of proof here is borrowed from [CT14].…”

Section: Generalized Okounkov Bodies On Complex Surfacesmentioning

confidence: 99%

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Transcendental Morse inequality and generalised Okounkov bodies

Deng

2017

Alg. Geom.

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The main goal of this article is to construct "arithmetic Okounkov bodies" for an arbitrary pseudo-effective (1,1)-class α on a Kähler manifold. First, we prove the differentiability of volumes of big classes for Kähler manifolds on which modified nef cones and nef cones coincide. As a consequence, we prove Demailly's transcendental Morse inequality for these particular Kähler manifolds. In the second part, we construct the generalized Okounkov body for any big (1,1)-class, and prove that it coincides with the Okounkov body when the big class is rational. Next, we give a complete characterization of generalized Okounkov bodies on surfaces, and relate the standard Euclidean volume of the body to the volume of the corresponding big class as defined by Boucksom; this solves a problem raised by Lazarsfeld and Mustaţȃ in the case of surfaces. Finally, we study the behavior of the generalized Okounkov bodies on the boundary of the big cones.

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Section: Generalized Okounkov Bodies On Complex Surfacesmentioning

confidence: 95%

Section: Ya Dengmentioning

confidence: 99%

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Transcendental Morse inequality and generalised Okounkov bodies

Deng

2017

Alg. Geom.

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“…We conclude this article by noting that a refinement of the extension and gluing techniques that we just presented allowed Collins and the author [15] to prove the following extension theorem for Kähler currents: It is expected that this extension result should hold more generally when V is an analytic subvariety and T is just a closed positive current in the class [ω| V ] (in which case T should extend to a global closed positive current). This was proved by Coman-Guedj-Zeriahi [18] when X is projective and [ω] is a rational class, using rather different techniques.…”

Section: Ideas From the Proofmentioning

confidence: 97%

Nakamaye’s theorem on complex manifolds

Tosatti¹

2018

Algebraic Geometry: Salt Lake City 2015

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“…• When ω is a Kähler metric and ϕ is a quasi-psh function on V, which has analytic singularities, such that ω| V + √ −1∂ ∂ϕ > ǫω| V for some ǫ > 0, there is a quasi-psh function Φ on X, such that Φ| V = ϕ and ω + √ −1∂ ∂Φ > ǫ ′ ω on X by Collins-Tosatti [CT14]. • When ω is a Kähler metric and ϕ is a quasi-psh function with arbitrary singularity on V, such that ω| V + √ −1∂ ∂ϕ > ǫω| V for some ǫ > 0.…”

Section: Introductionmentioning

confidence: 99%

On the extension of Kähler currents on compact Kähler manifolds: holomorphic retraction case

Ning¹,

Wang²,

Zhou³

2021

Preprint

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In the present paper, we show that given a compact Kähler manifold (X, ω) with a Kähler metric ω, and a complex submanifold V ⊂ X of positive dimension, if V has a holomorphic retraction structure in X, then any quasi-plurisubharmonic function ϕ on V such that ω| V + √ −1∂ ∂ϕ ≥ εω| V with ε > 0 can be extended to a quasi-plurisubharmonic function Φ on X, such that ω + √ −1∂ ∂Φ ≥ ε ′ ω for some ε ′ > 0. This is an improvement of results in [WZ20]. Examples satisfying the assumption that there exists a holomorphic retraction structure contain product manifolds, thus contains many compact Kähler manifolds which are not necessarily projective.

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An extension theorem for Kähler currents with analytic singularities

Abstract: Abstract. We prove an extension theorem for Kähler currents with analytic singularities in a Kähler class on a complex submanifold of a compact Kähler manifold.

Cited by 16 publications

References 17 publications

Transcendental Morse inequality and generalised Okounkov bodies

Transcendental Morse inequality and generalised Okounkov bodies

Nakamaye’s theorem on complex manifolds

On the extension of Kähler currents on compact Kähler manifolds: holomorphic retraction case

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