Polynomial convexity, rational convexity, and currents

Duval, Jean-Luc; Sibony, Nessim

doi:10.1215/s0012-7094-95-07912-5

Cited by 83 publications

(110 citation statements)

References 19 publications

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“…-Corollary 5.7 implies that T + can be approximated by currents of integration on vertical manifolds with control of support. When p = 1, this holds for every current in C v (D) [12]. The problem is still open for general currents of higher bidegree.…”

mentioning

confidence: 99%

Geometry of currents, intersection theory and dynamics of horizontal-like maps

Dinh¹,

Sibony²

2006

Ann. inst. Fourier

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We introduce a geometry on the cone of positive closed currents of bidegree (p, p) and apply it to define the intersection of such currents. We also construct and study the Green currents and the equilibrium measure for horizontal-like mappings. The Green currents satisfy some extremality properties. The equilibrium measure is invariant, mixing and has maximal entropy. It is equal to the intersection of the Green currents associated to the horizontal-like map and to its inverse.

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mentioning

confidence: 99%

Geometry of currents, intersection theory and dynamics of horizontal-like maps

Dinh¹,

Sibony²

2006

Ann. inst. Fourier

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“…Some of their properties are given in [21,1,2,13,9,14]. Given a compact Kähler manifold (X, ω) of dimension k, we want to define the intersection S ∧ T of a positive closed (1, 1)-current S with a positive pluriharmonic current T of bidegree (p, p), 1 ≤ p ≤ k − 1.…”

Section: Intersection Of Currentsmentioning

confidence: 99%

Regularization of currents and entropy

Dinh

Sibony

2004

Annales Scientifiques de l’École Normale Supérieure

106

163

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Abstract. Let T be a positive closed (p, p)-current on a compact Kähler manifold X. Then, there exist smooth positive closed (p, p)-forms T≤ c X T where c X > 0 is a constant independent of T . We also extend this result to positive pluriharmonic currents. Then we study the wedge product of positive closed (1, 1)-currents having continuous potential with positive pluriharmonic currents. As an application, we give an estimate for the topological entropy of meromorphic maps on compact Kähler manifolds.

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“…The second corollary is less general than Theorem 1 in that there are examples of compact, holomorphically convex, real-analytic subvarieties of C N that are not rationally convex: Duval and Sibony [7] have proved that there is no rationally convex three-sphere in C 3 . On the other hand, Ahern and Rudin [1] have shown how to embed a three-sphere in C 3 as a totally real manifold.…”

Section: Briefly X Is the Spectrum Of O O O(x) Or Speco O O(x) = Xmentioning

confidence: 99%