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.

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.

[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. | MR | Zbl

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

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

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

[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. | MR | Zbl

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

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

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

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

[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. | JFM | MR

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

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

[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. | MR | Zbl

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

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

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

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

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

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

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

[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. | MR | Zbl

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

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

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

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

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

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

[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. | MR | Zbl

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

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

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

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

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

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

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