[Plurisubharmonic functions in almost complex manifolds]
Let (M,J) be an almost complex manifold. An upper semi-continuous map u:(M,J)→[−∞,+∞[ is said to be plurisubharmonic if u∘ϕ is subharmonic for every pseudo-holomorphic curve: ϕ:(Δ,J0)→(M,J). By using regularization techniques for currents and Taylor series expansions in suitable coordinates with respect to the structure J, we prove that an upper semi-continuous map u:(M,J)→[−∞,+∞[ which is not identically equal to −∞ is plurisubharmonic if and only if the (1,1)-part of is (semi-)positive as a current.
Soit (M,J) une variété presque complexe. Une application u :(M,J)→[−∞,+∞[ semi-continue supérieurement est dite plurisousharmonique si u∘ϕ est sousharmonique, pour toute courbe pseudo-holomorphe ϕ :(Δ,J0)→(M,J). En utilisant des techniques de régularisation des courants et des développements de Taylor dans des coordonnées locales adaptées à la structure J, on démontre qu'une application u :(M,J)→[−∞,+∞[ semi-continue supérieurement et non identiquement égale à −∞ est plurisousharmonique si et seulement si la partie de type (1,1) de est (semi-)positive au sens des courants.
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Fathi Haggui 1
@article{CRMATH_2002__335_6_509_0, author = {Fathi Haggui}, title = {Fonctions {PSH} sur une vari\'et\'e presque complexe}, journal = {Comptes Rendus. Math\'ematique}, pages = {509--514}, publisher = {Elsevier}, volume = {335}, number = {6}, year = {2002}, doi = {10.1016/S1631-073X(02)02518-9}, language = {fr}, }
Fathi Haggui. Fonctions PSH sur une variété presque complexe. Comptes Rendus. Mathématique, Volume 335 (2002) no. 6, pp. 509-514. doi : 10.1016/S1631-073X(02)02518-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02518-9/
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