<i>m</i>-Potential theory associated to a positive closed current in the class of<i>m</i>-sh functions

Dhouib, Abir; Elkhadhra, Fredj

doi:10.1080/17476933.2015.1133615

Cited by 12 publications

(30 citation statements)

References 10 publications

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“…Proof. First of all by following verbatim the proofs given in [Prop 1.2.4 in [11]] and [Lemma 2 in [7]] we see that if u j is a sequence of smooth m-subharmonic functions converging to a locally bounded m-subharmonic function u then u j converges to u in cap V,m on each E ⊂⊂ D ( †). Since D is smoothly bounded we can assume that u is bounded in a neighborhood of D. For a natural approximation u j of u we know u j 's are m-subharmonic and converge uniformly on compacta to u by Proposition 2.8 so by ( †) for each k ∈ N, there exists j k such that…”

Section: Convergence Of M-positive Supercurrents and Quasicontinuity Of Msubharmonic Functionsmentioning

confidence: 94%

“…During this study we will follow the positivity definitions given by [7] so that we can use the arguments of standard positivity and guarantee that any m-positive form defines an m-positive current. Definition 2.5.…”

Section: Esmentioning

confidence: 99%

“…Later this important tool was used in the global theory of Monge-Ampère equation over the compact Kähler manifolds by [9] in connection with other complex variants as Alexander capacity and Tchebychev constants. In [7] the idea of capacity is generalized to the relative m-capacity where the capacity of each compact is calculated in association with an m-positive closed current. In this current work we are interested in understanding the relative capacity with respect to the closed m-positive supercurrents which are closely related to the intersection of tropical hypersurfaces in superspaces.…”

Section: Tropical Geometry and Relative Capacity With Respect To Tropical Varietiesmentioning

confidence: 99%

“…In this note we will extend these formal ideas further into another direction namely m-pluripotential theory which is introduced by Blocki to study the behaviour of complex Hessian equation. Later in [11] and [7] the ideas of Blocki were connected to m-positive currents. Now in this study we will take this connection into Riemannian superspace setting where m-positive closed currents actually give information about the intersection of tropical hypersurfaces.…”

Section: Introductionmentioning

confidence: 99%

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m-Pluripotential theory on Riemannian spaces and tropical geometry

Şahin

2021

Hacettepe Journal of Mathematics and Statistics

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In this study we extend the concepts of m-pluripotential theory to the Riemannian superspace formalism. Since in this setting positive supercurrents and tropical varieties are closely related, we try to understand the relative capacity notion with respect to the intersection of tropical hypersurfaces. Moreover, we generalize the classical quasicontinuity result of Cartan to m-subharmonic functions of Riemannian spaces and lastly we introduce the indicators of m-subharmonic functions and give a geometric characterization of their Newton numbers.

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Section: Convergence Of M-positive Supercurrents and Quasicontinuity Of Msubharmonic Functionsmentioning

confidence: 94%

Section: Esmentioning

confidence: 99%

Section: Tropical Geometry and Relative Capacity With Respect To Tropical Varietiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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m-Pluripotential theory on Riemannian spaces and tropical geometry

Şahin

2021

Hacettepe Journal of Mathematics and Statistics

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“…m-positivity. Building on the work of Douib-Elkhadhra [6] on the m-complex pluripotential theory, S ¸ahin [11] has introduced recently the following notions of m-positivity in the superformalism context :…”

Section: M-positivity and Superhessian Operatormentioning

confidence: 99%

Lelong-Jensen formula, Demailly-Lelong numbers and weighted degree of positive supercurrents

Elkhadhra¹,

Zahmoul²

2018

Preprint

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The goal of this work is to extend the concepts of generalized Lelong number of positive currents investigated by Skoda, Demailly and Ghiloufi in complex analysis, to positive supercurrents on the real superspace. We generalize then a result of Lagerberg when the super current is closed as well as a very recent result of Berndtsson for minimal supercurrents associated to submanifolds of R n . The main tool is a variant of the well-known Lelong-Jensen formula in the superformalism case. Moreover, we extend to our setting various interesting theorems in complex analysis as Demailly and Rashkovskii comparison theorems. Finally, we complete the work begun by Lagerberg on the degree and the direct image of positive closed supercurrents.

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m-Generalized Lelong Numbers and Capacity Associated to a Class of m-Positive Closed Currents

Elkhadhra

2018

Results Math

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m-Potential theory associated to a positive closed current in the class ofm-sh functions

Cited by 12 publications

References 10 publications

m-Pluripotential theory on Riemannian spaces and tropical geometry

m-Pluripotential theory on Riemannian spaces and tropical geometry

Lelong-Jensen formula, Demailly-Lelong numbers and weighted degree of positive supercurrents

m-Generalized Lelong Numbers and Capacity Associated to a Class of m-Positive Closed Currents

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