A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces

Viêt-Anh Nguyên[1]

  • [1] Max-Planck-Institut für Mathematik Vivatsgasse 7 D–53111 Bonn, Germany

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

  • Volume: 4, Issue: 2, page 219-254
  • ISSN: 0391-173X

Abstract

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Using recent development in Poletsky theory of discs, we prove the following result: Let X , Y be two complex manifolds, let Z be a complex analytic space which possesses the Hartogs extension property, let A (resp. B ) be a non locally pluripolar subset of X (resp. Y ). We show that every separately holomorphic mapping f : W : = ( A × Y ) ∪ ( X × B ) → Z extends to a holomorphic mapping f ^ on W ^ : = ( z , w ) ∈ X × Y : ω ˜ ( z , A , X ) + ω ˜ ( w , B , Y ) < 1 such that f ^ = f on W ∩ W ^ , where ω ˜ ( · , A , X ) (resp. ω ˜ ( · , B , Y ) ) is the plurisubharmonic measure of A (resp. B ) relative to X (resp. Y ). Generalizations of this result for an N -fold cross are also given.

How to cite

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Nguyên, Viêt-Anh. "A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.2 (2005): 219-254. <http://eudml.org/doc/84559>.

@article{Nguyên2005,
abstract = {Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$$Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f:\ W:=(A\times Y) \cup (X\times B)\rightarrow Z$ extends to a holomorphic mapping $\hat\{f\}$ on $\widehat\{W\}:=\left\lbrace (z,w)\in X\times Y:\ \widetilde\{\omega \}(z,A,X)+\widetilde\{\omega \}(w,B,Y)&lt;1 \right\rbrace $ such that $\hat\{f\}=f$ on $W\cap \widehat\{W\},$ where $\widetilde\{\omega \}(\cdot ,A,X)$ (resp. $\widetilde\{\omega \}(\cdot ,B,Y))$ is the plurisubharmonic measure of $A$ (resp. $B$) relative to $X$ (resp. $Y$). Generalizations of this result for an $N$-fold cross are also given.},
affiliation = {Max-Planck-Institut für Mathematik Vivatsgasse 7 D–53111 Bonn, Germany},
author = {Nguyên, Viêt-Anh},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {219-254},
publisher = {Scuola Normale Superiore, Pisa},
title = {A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces},
url = {http://eudml.org/doc/84559},
volume = {4},
year = {2005},
}

TY - JOUR
AU - Nguyên, Viêt-Anh
TI - A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2005
PB - Scuola Normale Superiore, Pisa
VL - 4
IS - 2
SP - 219
EP - 254
AB - Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$$Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f:\ W:=(A\times Y) \cup (X\times B)\rightarrow Z$ extends to a holomorphic mapping $\hat{f}$ on $\widehat{W}:=\left\lbrace (z,w)\in X\times Y:\ \widetilde{\omega }(z,A,X)+\widetilde{\omega }(w,B,Y)&lt;1 \right\rbrace $ such that $\hat{f}=f$ on $W\cap \widehat{W},$ where $\widetilde{\omega }(\cdot ,A,X)$ (resp. $\widetilde{\omega }(\cdot ,B,Y))$ is the plurisubharmonic measure of $A$ (resp. $B$) relative to $X$ (resp. $Y$). Generalizations of this result for an $N$-fold cross are also given.
LA - eng
UR - http://eudml.org/doc/84559
ER -

References

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  1. [1] O. Alehyane et J. M. HecartPropriété de stabilité de la fonction extrémale relative, preprint, (1999). MR2081144
  2. [2] 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. Zbl0237.32008MR352526
  3. [3] O. Alehyane et 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. Zbl0979.32011MR1841529
  4. [4] E. Bedford, The operator ( d d c ) n on complex spaces, Semin. P. Lelong - H. Skoda, Analyse, Années 1980/81, Lect. Notes Math. 919 (1982), 294–323. Zbl0479.32006MR658889
  5. [5] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1–40. Zbl0547.32012MR674165
  6. [6] A. Edigarian, Analytic discs method in complex analysis, Dissertationes Math. 402 (2002). Zbl0993.31003MR1897580
  7. [7] A. Edigarian and E. A. Poletsky, Product property of the relative extremal function, Bull. Polish Acad. Sci. Math. 45 (1997), 331–335. Zbl0898.32010MR1489875
  8. [8] F. Hartogs, Zur Theorie der analytischen Funktionen mehrer unabhängiger Veränder- lichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann. 62 (1906), 1–88. Zbl37.0444.01MR1511365JFM37.0444.01
  9. [9] B. Josefson, On the equivalence between polar and globally polar sets for plurisubharmonic functions on ℂ n , Ark. Mat. 16 (1978), 109–115. Zbl0383.31003MR590078
  10. [10] S. M. Ivashkovich, The Hartogs phenomenon for holomorphically convex Kähler manifolds, Math. USSR-Izv. 29 (1997), 225–232. Zbl0618.32011
  11. [11] M. Jarnicki and P. Pflug, “Extension of Holomorphic Functions”, de Gruyter Expositions in Mathematics 34, Walter de Gruyter, 2000. Zbl0976.32007MR1797263
  12. [12] M. Jarnicki and P. Pflug, An extension theorem for separately holomorphic functions with analytic singularities, Ann. Polon Math. 80 (2003), 143–161. Zbl1023.32001MR1972841
  13. [13] M. Jarnicki and P. Pflug, An extension theorem for separately holomorphic functions with pluripolar singularities, Trans. Amer. Math. Soc. 355 (2003), 1251–1267. Zbl1012.32002MR1938756
  14. [14] M. Jarnicki and P. Pflug, An extension theorem for separately meromorphic functions with pluripolar singularities, Kyushu J. of Math., 57 (2003), 291–302. Zbl1055.32002MR2050087
  15. [15] M. Klimek, “Pluripotential theory”, London Mathematical society monographs, Oxford Univ. Press., 6, 1991. Zbl0742.31001MR1150978
  16. [16] N. V. Khue and N. H. Thanh, Locally bounded holomorphic functions and the mixed Hartogs theorem, Southeast Asian Bull. Math. 23 (1999), 643–655. Zbl0948.32003MR1810829
  17. [17] F. Lárusson and R. Sigurdsson, Plurisubharmonic functions and analytic discs on manifolds, J. Reine Angew. Math. 501 (1998), 1–39. Zbl0901.31004MR1637837
  18. [18] Nguyên Thanh Vân, Separate analyticity and related subjects, Vietnam J. Math. 25 (1997), 81–90. Zbl0898.32002MR1681531
  19. [19] Nguyên Thanh Vân, Note on doubly orthogonal system of Bergman, In: “Linear Topological Spaces and Complex Analysis” 3 (1997), 157–159. Zbl0910.46009MR1632495
  20. [20] Ph. Noverraz, Fonctions plurisousharmoniques et analytiques dans les espaces vectoriels topologiques, Ann. Inst. Fourier Grenoble 19 (1969), 419–493. Zbl0176.09903MR265628
  21. [21] Nguyên Thanh Vân and J. Siciak, Fonctions plurisousharmoniques extrémales et systèmes doublement orthogonaux de fonctions analytiques, Bull. Sci. Math. 115 (1991), 235–244. Zbl0810.32011MR1101026
  22. [22] Nguyên Thanh Vân et A. Zeriahi, Familles de polynômes presque partout bornées, Bull. Sci. Math. 107 (1983), 81–89. Zbl0523.32011MR699992
  23. [23] Nguyên Thanh Vân et 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. Zbl0918.32001MR1228880
  24. [24] Nguyên Thanh Vân et A. Zeriahi, Systèmes doublement orthogonaux de fonctions holomorphes et applications, Banach Center Publ. 31, Inst. Math., Polish Acad. Sci. (1995), 281–297. Zbl0844.31003MR1341397
  25. [25] P. Pflug, Extension of separately holomorphic functions–a survey 1899–2001, Ann. Polon. Math. 80 (2003), 21–36. Zbl1023.32002MR1972831
  26. [26] 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. 52, Part 1 (1991), 163–171. Zbl0739.32015MR1128523
  27. [27] E. A. Poletsky, Holomorphic currents, Indiana Univ. Math. J. 42 (1993), 85–144. Zbl0811.32010MR1218708
  28. [28] P. Pflug and V.-A. Nguyên, A boundary cross theorem for separately holomorphic functions, Ann. Polon. Math. 84 (2004), 237–271. Zbl1068.32010MR2110930
  29. [29] P. Pflug and V.-A. Nguyên, Generalization of Drużkowski’s and Gonchar’s “Edge-of-the-Wedge” Theorems, preprint 2004, available at arXiv:math.CV/0503326. 
  30. [30] P. Pflug and V.-A. Nguyên, Envelope of holomorphy for boundary cross sets, preprint 2005. Zbl1137.32008MR2355152
  31. [31] T. Ransford, “Potential theory in the complex plane”, London Mathematical Society Student Texts 28, Cambridge: Univ. Press., 1995. Zbl0828.31001MR1334766
  32. [32] J. P. Rosay, Poletsky theory of disks on holomorphic manifolds, Indiana Univ. Math. J. 52 (2003), 157–169. Zbl1033.31006MR1970025
  33. [33] A. Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Russian Math. Surveys 36 (1981), 61–119. Zbl0494.31005MR629683
  34. [34] B. Shiffman, Extension of holomorphic maps into Hermitian manifolds, Math. Ann. 194 (1971), 249–258. Zbl0219.32007MR291507
  35. [35] I. Shimoda, Notes on the functions of two complex variables, J. Gakugei Tokushima Univ. 8 (1957), 1–3. Zbl0083.07003MR98844
  36. [36] J. Siciak, Analyticity and separate analyticity of functions defined on lower dimensional subsets of ℂ n , Zeszyty Nauk. Univ. Jagiello. Prace Mat. Zeszyt 13 (1969), 53–70. Zbl0285.32011MR247132
  37. [37] J. Siciak, Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of ℂ n , Ann. Polon. Math. 22 (1970), 145–171. Zbl0185.15202MR252675
  38. [38] T. Terada, Sur une certaine condition sous laquelle une fonction de plusieurs variables complexes est holomorphe, Publ. Res. Inst. Math. Sci. 2 (1967), 383–396. Zbl0183.34903MR240334
  39. [39] V. P. Zahariuta, Separately analytic functions, generalizations of the Hartogs theorem and envelopes of holomorphy, Math. USSR-Sb. 30 (1976), 51–67. Zbl0381.32003
  40. [40] A. Zeriahi, Comportement asymptotique des systèmes doublement orthogonaux de Bergman: Une approche élémentaire, Vietnam J. Math. 30 (2002), 177–188. Zbl1026.32010MR1934347
  41. [41] H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193–233. Zbl0158.33301MR224869

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