A Dolbeault Lemma for Temperate Currents

Skoda, Henri

doi:10.5802/ahl.92

Skoda, Henri

A Dolbeault Lemma for Temperate Currents

Annales Henri Lebesgue, Volume 4 (2021), pp. 879-896.

Metadata

DOI10.5802/ahl.92

Keywords Stein open subset of $\mathbb{C}^n$ or of a Stein manifold, $L^2$ estimates, $\bar{\partial }$-operator, Dolbeault $\bar{\partial }$-complex, temperate distributions and currents, temperate cohomology, Sobolev spaces

Abstract

We consider a bounded open Stein subset $Ω$ of a complex Stein manifold $X$ of dimension $n$ . We prove that if $f$ is a current on $X$ of bidegree $(p, q + 1)$ , $\bar{\partial}$ -closed on $Ω$ , we can find a current $u$ on $X$ of bidegree $(p, q)$ which is a solution of the equation $\bar{\partial} u = f$ in $Ω$ . In other words, we prove that the Dolbeault complex of temperate currents on $Ω$ (i.e. currents on $Ω$ which extend to currents on $X$ ) is concentrated in degree $0$ . Moreover if $f$ is a current on $X = ℂ^{n}$ of order $k$ , then we can find a solution $u$ which is a current on $ℂ^{n}$ of order $k + 2 n + 1$ .

References

[Dem82] Demailly, Jean-Pierre Estimations $L^{2}$ pour l’opérateur $\bar{\partial}$ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. Éc. Norm. Supér., Volume 15 (1982), pp. 457-511 | DOI | Numdam | Zbl

[Dem12] Demailly, Jean-Pierre Complex Analytic and Differential Geometry, 2012 (https://www-fourier.ujf-grenoble.fr/~Demailly/manuscripts/agbook.pdf, Open Content Book. Chapter VIII, paragraph 6, Theorem 6.9, p. 379)

[Dol56] Dolbeault, Pierre Formes différentielles et cohomologie sur une variété analytique complexe, Ann. Math., Volume 64 (1956), pp. 83-130 | DOI | MR | Zbl

[Dol57] Dolbeault, Pierre Formes différentielles et cohomologie sur une variété analytique complexe II, Ann. Math., Volume 65 (1957), pp. 282-330 | DOI | MR | Zbl

[Ele75] Elencwajg, Georges Pseudoconvexité locale dans les variétés kählériennes, Ann. Inst. Fourier, Volume 25 (1975) no. 2, pp. 295-314 | DOI | Numdam | MR | Zbl

[Hör65] Hörmander, Lars $L^{2}$ Estimates and Existence Theorem for the $\bar{\partial}$ Operator, Acta Math., Volume 113 (1965), pp. 89-152 | DOI | MR | Zbl

[Hör83] Hörmander, Lars The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften, 256, Springer, 1983 | Zbl

[Hör90] Hörmander, Lars An Introduction to Complex Analysis in Several Variables, North-Holland Mathematical Library, 7, North-Holland, 1990 | MR | Zbl

[KS96] Kashiwara, Masaki; Schapira, Pierre Moderate and Formal Cohomology associated with Constructibles Sheaves, Mém. Soc. Math. Fr., Nouv. Sér., Volume 64 (1996), pp. iii-64 | Numdam | Zbl

[Lel64] Lelong, Pierre Fonctions entières (n variables) et fonctions plurisousharmoniques d’ordre fini dans $ℂ^{n}$ , J. Anal. Math., Volume 12 (1964), pp. 365-406 | DOI | MR | Zbl

[Sch50] Schwartz, Laurent Théorie des distributions. Tome I, Actualités Scientifiques Industrielles ; Publications de l’Institut de Mathématique de l’Université de Strasbourg, 1091-9, Hermann, 1950 (2ème édition en 1957, réédition, 1966) | Zbl

[Sch17] Schapira, Pierre Microlocal Analysis and beyond (2017) (https://arxiv.org/abs/1701.08955)

[Sch21] Schapira, Pierre Vanishing of Temperate Cohomology on Complex Manifolds, Ann. Henri Lebesgue, Volume 4 (2021), pp. 863-877 | DOI

[Siu70] Siu, Yum-Tong Holomorphic Functions of Polynomial Growth on Bounded Domains, Duke Math. J., Volume 37 (1970), pp. 77-84 | MR | Zbl

[Sko71] Skoda, Henri $d^{''}$ -cohomologie à croissance lente dans C $^{n}$ , Ann. Sci. Éc. Norm. Supér., Volume 4 (1971) no. 1, pp. 97-120 | DOI | MR | Zbl

[Sko20] Skoda, Henri A Dolbeault lemma for temperate currents (2020) (https://arxiv.org/abs/2003.11437)

[Wei58] Weil, André Introduction à l’étude des variétés kählériennes, Actualités Scientifiques et Industrielles, 1267, Hermann, 1958 | Zbl

Metadata

Abstract

References

Cited by