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Cross theorems with singularities. (English) Zbl 1189.32006

The authors present a new approach to the extension of separately holomorphic functions defined on boundary crosses with singularities. This approach is based on the Poletsky theory of holomorphic discs and it permits to get cross theorems under weak assumptions.
The main result of the paper is the following extension theorem. Let \(X, Y\) be countable at infinity complex manifolds and let \(Z\) be a countable at infinity reduced complex analytic space with the Hartogs extension property. Let \(D\subset X\), \(G\subset Y\) be open, let \(A\subset\overline D\), \(B\subset\overline G\), \(A^\partial:=A\cap\partial D\), \(B^\partial:=B\cap\partial G\), \(W:=((D\cup A)\times B)\cup(A\times(G\cup B))\), \(W^o:=(D\times B)\cup(A\times G)\subset W\), \(\widetilde W:=(\widetilde A\times G)\cup(D\times\widetilde B)\), \(\widehat{\widetilde W}:=\{(z,w)\in D\times G: \omega(z,\widetilde A,D)+\omega(w,\widetilde B,G)<1\}\), where the sets \(\widetilde A\subset\overline D\), \(\widetilde B\subset\overline D\) and the special relative extremal functions \(\omega(\cdot,\widetilde A,D)\), \(\omega(\cdot,\widetilde B,G)\) are defined in the paper. Let \(M\subset W\) be a relatively closed set such that \(M\cap((A^\partial\times B)\cup(A\times B^\partial))=\emptyset\) and for every \((a,b)\in A\times B\) the fibers \(M_{(a,\cdot)}:=\{w\in G: (a,w)\in M\}\), \(M_{(\cdot,b)}:=\{z\in D: (z,b)\in M\}\) are thin (resp., locally pluripolar). Then there exists a relatively closed analytic (resp., locally pluripolar) set \(\widehat M\subset\widehat{\widetilde W}\) with \(\widehat M\cap\widetilde W\subset M\) such that for every separately continuous function \(f:W\setminus M\longrightarrow Z\) that is separately holomorphic on \(W^o\setminus M\), bounded along \(((A^\partial\times G)\cup(D\times B^\partial))\setminus M\), and such that \(f|_{(A\times B)\setminus M}\) is continuous at \(A^\partial\times B^\partial\), there exists a function \(\widehat f\in\mathcal O(\widehat{\widetilde W}\setminus\widehat M, Z)\) such that \(\lim_{(z,w)\to(\zeta,\eta)}\widehat f(z,w)=f(\zeta,\eta)\) (the limit taken in a special sense defined in the paper) for every \((\zeta,\eta)\in\widetilde W\setminus M\).

MSC:

32D15 Continuation of analytic objects in several complex variables
32D10 Envelopes of holomorphy
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References:

[1] Alehyane, O., Zeriahi, A.: Une nouvelle version du théorème d’extension de Hartogs pour les applications séparément holomorphes entre espaces analytiques. Ann. Pol. Math. 76, 245–278 (2001) · Zbl 0979.32011 · doi:10.4064/ap76-3-5
[2] Alehyane, O., Hecart, J.M.: Propriété de stabilité de la fonction extrémale relative. Potential Anal. 21(4), 363–373 (2004) · Zbl 1064.32024 · doi:10.1023/B:POTA.0000034326.53466.fe
[3] Bedford, E.: The operator (dd c ) n on complex spaces. In: Semin. P. Lelong–H. Skoda, Analyse, Années 1980/81. Lect. Notes Math., vol. 919, pp. 294–323. Springer, Berlin (1982)
[4] Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982) · Zbl 0547.32012 · doi:10.1007/BF02392348
[5] Chirka, E.M.: The extension of pluripolar singularity sets. Proc. Steklov Inst. Math. 200, 369–373 (1993) · Zbl 0794.32011
[6] Coupet, B.: Construction de disques analytiques et régularité de fonctions holomorphes au bord. Math. Z. 209(2), 179–204 (1992) · Zbl 0787.32028 · doi:10.1007/BF02570829
[7] Ivashkovich, S.M.: The Hartogs phenomenon for holomorphically convex Kähler manifolds. Math. USSR Izv. 29, 225–232 (1997) · Zbl 0618.32011 · doi:10.1070/IM1987v029n01ABEH000968
[8] Jarnicki, M., Pflug, P.: Extension of Holomorphic Functions. De Gruyter Expositions in Mathematics, vol. 34. de Gruyter, Berlin (2000) · Zbl 0976.32007
[9] Jarnicki, M., Pflug, P.: An extension theorem for separately holomorphic functions with analytic singularities. Ann. Pol. Math. 80, 143–161 (2003) · Zbl 1023.32001 · doi:10.4064/ap80-0-12
[10] Jarnicki, M., Pflug, P.: An extension theorem for separately holomorphic functions with pluripolar singularities. Trans. Am. Math. Soc. 355(3), 1251–1267 (2003) · Zbl 1012.32002 · doi:10.1090/S0002-9947-02-03193-8
[11] Jarnicki, M., Pflug, P.: An extension theorem for separately meromorphic functions with pluripolar singularities. Kyushu J. Math. 57(2), 291–302 (2003) · Zbl 1055.32002 · doi:10.2206/kyushujm.57.291
[12] Jarnicki, M., Pflug, P.: A remark on separate holomorphy. Stud. Math. 174(3), 309–317 (2006) · Zbl 1096.32005 · doi:10.4064/sm174-3-5
[13] Jarnicki, M., Pflug, P.: A general cross theorem with singularities. Analysis (Munich) 27(2–3), 181–212 (2007) · Zbl 1147.32012
[14] Josefson, B.: On the equivalence between polar and globally polar sets for plurisubharmonic functions on \(\mathbb{C}\) n . Ark. Mat. 16, 109–115 (1978) · Zbl 0383.31003 · doi:10.1007/BF02385986
[15] Klimek, M.: Pluripotential Theory. London Mathematical Society Monographs, vol. 6. Oxford University Press, London (1991) · Zbl 0742.31001
[16] Nguyên, V.-A.: A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci., Ser. V IV(2), 219–254 (2005) · Zbl 1170.32306
[17] Nguyên, V.-A.: A unified approach to the theory of separately holomorphic mappings. Ann. Sc. Norm. Super. Pisa Cl. Sci., Ser. V VII(2), 181–240 (2008) · Zbl 1241.32008
[18] Nguyên, V.-A.: Conical plurisubharmonic measure and new cross theorems. arXiv:0901.3222 [math.CV] · Zbl 1188.32009
[19] Nguyên, V.-A.: Recent developments in the theory of separately holomorphic mappings. Colloq. Math. (to appear). arXiv:0901.1991 [math.CV], 32 pp. · Zbl 1189.32005
[20] Nguyên, V.-A., Pflug, P.: Boundary cross theorem in dimension 1 with singularities. Indiana Univ. Math. J. 58(1), 393–414 (2009) · Zbl 1171.32005 · doi:10.1512/iumj.2009.58.3478
[21] Pflug, P.: Extension of separately holomorphic functions–a survey 1899–2001. Ann. Pol. Math. 80, 21–36 (2003) · Zbl 1023.32002 · doi:10.4064/ap80-0-1
[22] Pflug, P., Nguyên, V.-A.: A boundary cross theorem for separately holomorphic functions. Ann. Pol. Math. 84, 237–271 (2004) · Zbl 1068.32010 · doi:10.4064/ap84-3-6
[23] Pflug, P., Nguyên, V.-A.: Boundary cross theorem in dimension 1. Ann. Pol. Math. 90(2), 149–192 (2007) · Zbl 1122.32006 · doi:10.4064/ap90-2-5
[24] Pflug, P., Nguyên, V.-A.: Generalization of a theorem of Gonchar. Ark. Mat. 45, 105–122 (2007) · Zbl 1161.31005 · doi:10.1007/s11512-006-0025-6
[25] Pflug, P., Nguyên, V.-A.: Envelope of holomorphy for boundary cross sets. Arch. Math. (Basel) 89, 326–338 (2007) · Zbl 1137.32008
[26] Poletsky, E.A.: Plurisubharmonic functions as solutions of variational problems. In: Several Complex Variables and Complex Geometry, vol. 1, Proc. Summer Res. Inst., Santa Cruz/CA (USA), 1989. Proc. Symp. Pure Math., vol. 52, pp. 163–171. Am. Math. Soc., Providence (1991) · Zbl 0739.32015
[27] Poletsky, E.A.: Holomorphic currents. Indiana Univ. Math. J. 42(1), 85–144 (1993) · Zbl 0811.32010 · doi:10.1512/iumj.1993.42.42006
[28] Rosay, J.P.: Poletsky theory of disks on holomorphic manifolds. Indiana Univ. Math. J. 52(1), 157–169 (2003) · Zbl 1033.31006 · doi:10.1512/iumj.2003.52.2170
[29] Shiffman, B.: Extension of holomorphic maps into Hermitian manifolds. Math. Ann. 194, 249–258 (1971) · Zbl 0219.32007 · doi:10.1007/BF01350128
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