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Pluripolar graphs are holomorphic. (English) Zbl 1114.32001

The main result asserts that let \(\Omega\) be a domain in \(\mathbb{C}^n\) and let \(f: \Omega \rightarrow \mathbb{C}\) be a continuous function, then the graph \(\Gamma(f)\) of \(f\) is a pluripolar subset of \(\mathbb{C}^{n+1}\) if and only if \(f\) is holomorphic.

MSC:

32A10 Holomorphic functions of several complex variables
32U05 Plurisubharmonic functions and generalizations
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References:

[1] Alexander, H., Linking and holomorphic hulls.J. Differential Geom., 38 (1993), 151–160. · Zbl 0792.32010
[2] Browder, A., Cohomology of maximal ideal spaces.Bull. Amer. Math. Soc., 67 (1961), 515–516. · Zbl 0107.09501 · doi:10.1090/S0002-9904-1961-10663-0
[3] Chirka, E. M.,Complex Analytic Sets. ”Nauka”, Moscow, 1985 (Russian); English translation: Mathematics and its Applications (Soviet Series), 46. Kluwer, Dordrecht, 1989. · Zbl 0781.32011
[4] Chirka, E. M. &Henkin, G. M., Boundary properties of holomorphic functions of several complex variables, inCurrent Problems in Mathematics, 4, pp. 12–142. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1975 (Russian). · Zbl 0335.32001
[5] Edigarian, A., Graphs of multifunctions.Math. Z., 250 (2005), 145–147. · Zbl 1089.32031 · doi:10.1007/s00209-004-0746-9
[6] Hörmander, L.,An Introduction to Complex Analysis in Several, Variables, 3rd edition. North-Holland Mathematical Library, 7. North-Holland, Amsterdam, 1990. · Zbl 0685.32001
[7] Josefson, B., On the equivalence between locally polar and globally polar sets for plurisubharmonic functions onC n .Ark. Mat., 16 (1978), 109–115. · Zbl 0383.31003 · doi:10.1007/BF02385986
[8] Nishino, T., Sur les valeurs exceptionnelles au sens de Picard d’une fonction entière de deux variables.J. Math. Kyoto Univ., 2 (1962/63), 365–372. · Zbl 0146.10801
[9] –,Function Theory in Several Complex Variables. Translated from the 1996 Japanese original. Transl. Math. Monographs, 193. Amer. Math. Soc., Providence, RI, 2001.
[10] Ohsawa, T., Analyticity of complements of complete Kähler domains.Proc. Japan Acad. Ser. A Math. Sci., 56 (1980), 484–487. · Zbl 0485.32006 · doi:10.3792/pjaa.56.484
[11] Shcherbina, N., Pluripolar multifunctions are analytic. Preprint, 2003.
[12] Spanier, E. H.,Algebraic Topology. McGraw-Hill, New York-Toronto-London, 1966.
[13] Tsuji, M.,Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. · Zbl 0087.28401
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