A cut-off theorem for plurisubharmonic currents

Bassanelli, Giovanni

doi:10.1515/form.1994.6.567

Cited by 39 publications

(73 citation statements)

References 2 publications

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“…The reader can find others properties of theses classes of currents in Demailly [13], Skoda [34], Sibony, Berndtsson, Fornaess [31], [9], [17], Alessandrini and Bassanelli [2,3,8], etc. If T is a positive plurisubharmonic current, one can define a density ν(T, a) of T at every point a.…”

Section: Let V Be a Complex Manifold Of Dimension N A Current T Of Bmentioning

confidence: 99%

Polynomial hulls and positive currents

Dinh¹,

Lawrence²

2003

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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Let T be a positive plurisubharmonic current of bidimension (p, p) and let δ > 0. Assume that the Lelong number of T satisfies ν(T, a) ≥ δ on a dense subset of supp(T ) (rectifiable currents satisfy this condition). Then T = ϕ[X], where X is a complex subvariety of pure dimension p and ϕ is a weakly plurisubharmonic function on X. We have an analogous result for plurisuperharmonic currents. We also introduce and study a notion of polynomial p-hull.

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Section: Let V Be a Complex Manifold Of Dimension N A Current T Of Bmentioning

confidence: 99%

Polynomial hulls and positive currents

Dinh¹,

Lawrence²

2003

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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“…Hence, every limit value of (dd c ϕ n + Θ ′ ) ∧ τ * (S) is a positive dd c -closed current supported in ∆. Following Bassanelli [2], it is a current on ∆ (this is true for every positive current T supported in ∆ such that dd c T is of order 0). Hence, in order to prove (11) we only have to check that …”

Section: Pluriharmonic Currentsmentioning

confidence: 90%

Regularization of currents and entropy

Dinh

Sibony

2004

Annales Scientifiques de l’École Normale Supérieure

107

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 following result is the complex version of the classical support theorem in the real setting, [4,83,64]. The last property holds also when Y is a singular analytic set.…”

Section: A2 Positive Currents and Psh Functionsmentioning

confidence: 93%

Dynamics in Several Complex Variables

Abate

2017

Lecture Notes in Mathematics

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The emphasis of this introductory course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic (p.s.h.) functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. Nervertheless, we choose to show their effectiveness and to describe the theory for two large families of maps: the endomorphisms of projective spaces and the polynomial-like mappings.The first chapter deals with holomorphic endomorphisms of the projective space P k . We establish the first properties and give several constructions for the Green currents T p and the equilibrium measure µ = T k . The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems. We show the existence of a proper algebraic set E , totally invariant, i.e. f −1 (E ) = f (E ) = E , such that when a ∈ E , the probability measures, equidistributed on the fibers f −n (a), converge towards the equilibrium measure µ, as n goes to infinity. A similar result holds for the restriction of f to invariant subvarieties. We survey the equidistribution problem when points are replaced by varieties of arbitrary dimension, and discuss the equidistribution of periodic points. We then establish ergodic properties of µ: Kmixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. We heavily use the compactness of the space DSH(P k ) of differences of quasi-p.s.h. functions. In particular, we show that the measure µ is moderate, i.e. µ, e α|ϕ| ≤ c, on bounded sets of ϕ in DSH(P k ), for suitable positive constants α, c. Finally, we study the entropy, the Lyapounov exponents and the dimension of µ.The second chapter develops the theory of polynomial-like maps, i.e. proper holomorphic maps f : U → V where U, V are open subsets of C k with V convex and U ⋐ V . We introduce the dynamical degrees for such maps and construct the equilibrium measure µ of maximal entropy. Then, under a natural assumption on the dynamical degrees, we prove equidistribution properties of points and various statistical properties of the measure µ. The assumption is stable under small pertubations on the map. We also study the dimension of µ, the Lyapounov exponents and their variation.Our aim is to get a self-contained text that requires only a minimal background. In order to help the reader, an appendix gives the basics on p.s.h. functions, positive closed currents and super-potentials on projective spaces. Some exercises are proposed and an extensive bibliography is given.

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A cut-off theorem for plurisubharmonic currents

Cited by 39 publications

References 2 publications

Polynomial hulls and positive currents

Polynomial hulls and positive currents

Regularization of currents and entropy

Dynamics in Several Complex Variables

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