A unified approach to the theory of separately holomorphic mappings

Nguyên, Viêt-Anh 

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 2, pp. 181-240.

Résumé

We extend the theory of separately holomorphic mappings between complex analytic spaces. Our method is based on Poletsky theory of discs, Rosay theorem on holomorphic discs and our recent joint-work with Pflug on boundary cross theorems in dimension $1.$ It also relies on our new technique of conformal mappings and a generalization of Siciak’s relative extremal function. Our approach illustrates the unified character: “From local information to global extensions”. Moreover, it avoids systematically the use of the classical method of doubly orthogonal bases of Bergman type.

Détail

Classification : 32D15, 32D10

Bibliographie

[1] R. A. Airapetyan and G. M. Henkin, Analytic continuation of CR-functions across the “edge of the wedge", Dokl. Akad. Nauk SSSR 259 (1981), 777-781 (Russian). English transl.: Soviet Math. Dokl. 24 (1981), 128-132. | MR

[2] R. A. Airapetyan and G. M. Henkin, Integral representations of differential forms on Cauchy-Riemann manifolds and the theory of CR-functions. II, Mat. Sb. 127 (1985), 92-112, (Russian). English transl.: Math. USSR-Sb. 55 (1986), 99-111. | Zbl | MR

[3] O. Alehyane and J. M. Hecart, Propriété de stabilité de la fonction extrémale relative, Potential Anal. 21 (2004), 363-373. | Zbl | MR

[4] K. Adachi, M. Suzuki and M. Yoshida, Continuation of holomorphic mappings with values in a complex Lie group, Pacific J. Math. 47 (1973), 1-4. | Zbl | MR

[5] O. Alehyane and A. Zeriahi, Une nouvelle version du théorème d'extension de Hartogs pour les applications séparément holomorphes entre espaces analytiques, Ann. Polon. Math. 76 (2001), 245-278. | Zbl | MR

[6] E. Bedford, “The operator ${\left(d{d}^c\right)}^n$ on Complex Spaces”, Semin. P. Lelong - H. Skoda, Analyse, Années 1980/81, Lect. Notes Math. 919 (1982), 294-323. | Zbl | MR

[7] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40. | Zbl | MR

[8] S. Bernstein, “Sur l'Ordre de la Meilleure Approximation des Fonctions Continues par des Polynômes de Degré Donné”, Bruxelles, 1912. | JFM

[9] L. M. Drużkowski, A generalization of the Malgrange-Zerner theorem, Ann. Polon. Math. 38 (1980), 181-186. | Zbl | MR

[10] A. Edigarian, Analytic discs method in complex analysis, Diss. Math. 402 (2002), 56. | Zbl | MR

[11] G. M. Goluzin, “ Geometric Theory of Functions of a Complex Variable”, (English), Providence, R. I.: American Mathematical Society (AMS). VI, 1969, p. 676. | Zbl | MR

[12] A. A. Gonchar, On analytic continuation from the “edge of the wedge" theorem, Ann. Acad. Sci. Fenn. Math. Diss. 10 (1985), 221-225. | Zbl | MR

[13] A. A. Gonchar, On Bogolyubov's “edge-of-the-wedge" theorem, Proc. Steklov Inst. Math. 228 (2000), 18-24. | Zbl | MR

[14] F. Hartogs, Zur Theorie der analytischen Funktionen mehrer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann. 62 (1906), 1-88. | MR | JFM

[15] B. Jöricke, The two-constants theorem for functions of several complex variables, (Russian), Math. Nachr. 107 (1982), 17-52. | Zbl | MR

[16] B. Josefson, On the equivalence between polar and globally polar sets for plurisubharmonic functions on ${ℂ}^n$, Ark. Mat. 16 (1978), 109-115. | Zbl | MR

[17] S. M. Ivashkovich, The Hartogs phenomenon for holomorphically convex Kähler manifolds, Math. USSR-Izv. 29 (1997), 225-232. | Zbl

[18] M. Jarnicki and P. Pflug, “Extension of Holomorphic Functions”, de Gruyter Expositions in Mathematics n. 34, Walter de Gruyter, 2000. | Zbl | MR

[19] M. Jarnicki and P. Pflug, Invariant distances and metrics in complex analysis-revisited, Dissertationes Math. (Rozprawy Math.) 430 (2005). | Zbl | MR

[20] M. Klimek, “Pluripotential Theory”, London Mathematical society monographs, Oxford Univ. Press., n. 6, 1991. | Zbl | MR

[21] H. Komatsu, A local version of Bochner's tube theorem, J. Fac. Sci., Univ. Tokyo, Sect. I A 19 (1972), 201-214. | Zbl | MR

[22] F. Lárusson and R. Sigurdsson, Plurisubharmonic functions and analytic discs on manifolds, J. Reine Angew. Math. 501 (1998), 1-39. | Zbl | MR

[23] Nguyên Thanh Vân, Separate analyticity and related subjects, Vietnam J. Math. 25 (1997), 81-90. | Zbl | MR

[24] Nguyên Thanh Vân, Note on doubly orthogonal system of Bergman, Linear Topological Spaces and Complex Analysis 3 (1997), 157-159. | Zbl | MR

[25] Nguyên Thanh Vân and A. Zeriahi, Familles de polynômes presque partout bornées, Bull. Sci. Math. 107 (1983), 81-89. | Zbl | MR

[26] Nguyên Thanh Vân and A. Zeriahi, Une extension du théorème de Hartogs sur les fonctions séparément analytiques, In: “ Analyse Complexe Multivariable, Récents Développements”, A. Meril (ed.), EditEl, Rende, 1991, 183-194. | Zbl | MR

[27] Nguyên Thanh Vân and A. Zeriahi, Systèmes doublement orthogonaux de fonctions holomorphes et applications, Banach Center Publ. 31 (1995), 281-297. | Zbl | MR

[28] V.-A. Nguyên, A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (V) 4 (2005), 219-254. | Zbl | MR | mathdoc-id

[29] V.-A. Nguyên, Conical plurisubharmonic measure and new cross theorems, in preparation. | Zbl

[30] V.-A. Nguyên, Recent developments in the theory of separately holomorphic mappings, http://publications.ictp.it, IC/2007/074, 28 pages. | Zbl

[31] V.-A. Nguyên and P. Pflug, Boundary cross theorem in dimension $1$ with singularities, Indiana Univ. Math. J., to appear. | Zbl | MR

[32] V.-A. Nguyên and P. Pflug, Cross theorems with singularities. http://publications.ictp.it, IC/2007/073. | Zbl

[33] P. Pflug, Extension of separately holomorphic functions-a survey 1899-2001, Ann. Polon. Math. 80 (2003), 21-36. | Zbl | MR

[34] P. Pflug and V.-A. Nguyên, A boundary cross theorem for separately holomorphic functions, Ann. Polon. Math. 84 (2004), 237-271. | Zbl | MR

[35] P. Pflug and V.-A. Nguyên, Boundary cross theorem in dimension $1$, Ann. Polon. Math. 90 (2007), 149-192. | Zbl | MR

[36] P. Pflug and V.-A. Nguyên, Generalization of a theorem of Gonchar, Ark. Mat. 45 (2007), 105-122. | Zbl | MR

[37] P. Pflug and V.-A. Nguyên, Envelope of holomorphy for boundary cross sets, Arch. Math. (Basel) 89 (2007), 326-338. | Zbl | MR

[38] E. A. Poletsky, Plurisubharmonic functions as solutions of variational problems, In: “Several Complex Variables and Complex Geometry”, Proc. Summer Res. Inst., Santa Cruz/CA (USA) 1989, Proc. Symp. Pure Math. Vol. 52, Part 1, 1991, 163-171. | Zbl | MR

[39] E. A. Poletsky, Holomorphic currents, Indiana Univ. Math. J. 42 (1993), 85-144. | Zbl | MR

[40] T. Ransford, “Potential Theory in the Complex Plane”, London Mathematical Society Student Texts, n. 28, Cambridge: Univ. Press., 1995. | Zbl | MR

[41] J. P. Rosay, Poletsky theory of disks on holomorphic manifolds, Indiana Univ. Math. J. 52 (2003), 157-169. | Zbl | MR

[42] B. Shiffman, Extension of holomorphic maps into Hermitian manifolds, Math. Ann. 194 (1971), 249-258. | Zbl | MR

[43] B. Shiffman, Hartogs theorems for separately holomorphic mappings into complex spaces, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 89-94. | Zbl | MR

[44] J. Siciak, Analyticity and separate analyticity of functions defined on lower dimensional subsets of ${ℂ}^n$, Zeszyty Nauk. Univ. Jagiellon. Prace Mat. 13 (1969), 53-70. | Zbl | MR

[45] J. Siciak, Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of ${ℂ}^n$, Ann. Polon. Math. 22 (1970), 145-171. | Zbl | MR

[46] V. P. Zahariuta, Separately analytic functions, generalizations of the Hartogs theorem and envelopes of holomorphy, Math. USSR-Sb. 30 (1976), 51-67. | Zbl

[47] M. Zerner, Quelques résultats sur le prolongement analytique des fonctions de variables complexes, Séminaire de Physique Mathématique.

[48] A. Zeriahi, Comportement asymptotique des systèmes doublement orthogonaux de Bergman: Une approche élémentaire, Vietnam J. Math. 30 (2002), 177-188. | Zbl | MR

[49] H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193-233. | Zbl | MR