Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-24T12:13:42.326Z Has data issue: false hasContentIssue false
  • English
  • Français

Normal families of meromorphic mappings of several complex variables for moving hypersurfaces in a complex projective space

Part of: Holomorphic mappings and correspondences Holomorphic functions of several complex variables

Published online by Cambridge University Press:  11 January 2016

Gerd Dethloff ,
Do Duc Thai  and
Pham Nguyen Thu Trang
Gerd Dethloff
Affiliation:
Université Européenne de Bretagne, France, Université de Brest, Laboratoire de Mathématiques Bretagne Atlantique - UMR CNRS 6205, 29238 Brest Cedex 3, France, gerd.dethloff@univ-brest.fr
Do Duc Thai
Affiliation:
Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam, ducthai.do@gmail.com
Pham Nguyen Thu Trang
Affiliation:
Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam, pnttrangsp@yahoo.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The main aim of this article is to give sufficient conditions for a family of meromorphic mappings of a domain D in ℂn into ℙN(ℂ) to be meromorphically normal if they satisfy only some very weak conditions with respect to moving hypersurfaces in ℙN(ℂ), namely, that their intersections with these moving hypersurfaces, which moreover may depend on the meromorphic maps, are in some sense uniform. Our results generalize and complete previous results in this area, especially the works of Fujimoto, Tu, Tu-Li, Mai-Thai-Trang, and the recent work of Quang-Tan.

MSC classification

Secondary: 32A19: Normal families of functions, mappings 32H30: Value distribution theory in higher dimensions
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 217 , March 2015 , pp. 23 - 59
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Aladro, G. and Krantz, S. G., A criterion for normality in ℂn , J. Math. Anal. Appl. 161 (1991), 1–8. MR 1127544. DO 10.1016/0022-247X(91)90356-5.Google Scholar
[2] Chirka, E. M., Complex Analytic Sets, Math. Appl. (Soviet Ser.) 46, Kluwer, Dordrecht, 1989. MR 1111477. DOI 10.1007/978-94-009-2366-9.Google Scholar
[3] Eremenko, A., A Picard type theorem for holomorphic curves, Period. Math. Hungar. 38 (1999), 39–42. MR 1721476. DOI 10.1023/A:1004794914744.Google Scholar
[4] Fujimoto, H., On families of meromorphic maps into the complex projective space, Nagoya Math. J. 54 (1974), 21–51. MR 0367301.Google Scholar
[5] Joseph, J. E. and Kwack, M. H., Some classical theorems and families of normal maps in several complex variables, Complex Var. Theory Appl. 29 (1996), 343–362. MR 1390619.Google Scholar
[6] Joseph, J. E. and Kwack, M. H. Extension and convergence theorems for families of normal maps in several complex variables, Proc. Amer. Math. Soc. 125 (1997), 1675–1684. MR 1423310. DOI 10.1090/S0002-9939-97-04117-8.Google Scholar
[7] Lehto, O. and Virtanen, K. I., Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47–65. MR 0087746.Google Scholar
[8] Mai, P. N., Thai, D. D., and Trang, P. N. T., Normal families of meromorphic mappings of several complex variables into ℙ N (ℂ), Nagoya Math. J. 180 (2005), 91–110. MR 2186670.Google Scholar
[9] Nochka, E. I., On the theory of meromorphic functions(in Russian), Dokl. Akad. Nauk SSSR 269, no. 3 (1983), 547–552; English translation in Soviet Math. Dokl. 27, no. 2 (1983), 377-381. MR 0701289.Google Scholar
[10] Noguchi, J. and Winkelmann, J., Nevanlinna Theory in Several Complex Variables and Diophantine Approximation, Grundlehren Math. Wiss. 350, Springer, Tokyo, 2014. MR 3156076.Google Scholar
[11] Quang, S. D. and Tan, T. V., Normal families of meromorphic mappings of several complex variables into ℂPN for moving hyper surf aces, Ann. Polon. Math. 94 (2008), 97–110. MR 2438852. DOI 10.4064/ap94-2-1.Google Scholar
[12] Ru, M. and Stoll, W., ‘The Cartan conjecture for moving targets’ in Several Complex Variables and Complex Geometry, Part 2 (Santa Cruz, Calif., 1989), Proc. Sympos. Pure Math. 52, Amer. Math. Soc., Providence, 1991, 477-508. MR 1128565.Google Scholar
[13] Ru, M. and Stoll, W., The second main theorem for moving targets,, J. Geom. Anal. 1 (1991), 99–138. MR 1113373. DOI 10.1007/BF02938116.Google Scholar
[14] Rutishauser, H., Über Folgen und Scharen von analytischen und meromorphen Funk-tionen mehrerer Variabeln, sowie von analytischen Abbildungen, Acta Math. 83 (1950), 249–325. MR 0036322.CrossRefGoogle Scholar
[15] Stoll, W., Normal families of non-negative divisors, Math. Z. 84 (1964), 154–218. MR 0165142.CrossRefGoogle Scholar
[16] Stoll, W., Value Distribution Theory for Meromorphic Maps, Aspects Math. E7, Friedr. Vieweg, Braunschweig, 1985. MR 0823236. DOI 10.1007/978-3-663-05292-0.Google Scholar
[17] Thai, D. D. and Quang, S. D., Uniqueness problem with truncated multiplicities of meromorphic mappings in several complex variables for moving targets, Internat. J. Math. 16 (2005), 903–939. MR 2168074. DOI 10.1142/S0129167X05003132.Google Scholar
[18] Thai, D. D. and Quang, S. D., Second main theorem with truncated counting function in several complex variables for moving targets, Forum Math. 20 (2008), 163–179. MR 2386785. DOI 10.1515/FORUM.2008.007. CrossRefGoogle Scholar
[19] Thai, D. D., Trang, P. N. T., and Huong, P. D., Families of normal maps in several complex variables and hyperbolicity of complex spaces, Complex Var. Theory Appl. 48 (2003), 469–482. MR 1979525. DOI 10.1080/0278107031000094963.Google Scholar
[20] Tu, Z., Normality criterions for families of holomorphic mappings of several complex variables into PN(C), Proc. Amer. Math. Soc. 127 (1999), 1039–1049. MR 1469438. DOI 10.1090/S0002-9939-99-04610-9.Google Scholar
[21] Tu, Z., On meromorphically normal families of meromorphic mappings of several complex variables into PN(C), J. Math. Anal. Appl. 267 (2002), 1–19. MR 1886812. DOI 10.1006/jmaa.2000.6770.CrossRefGoogle Scholar
[22] Tu, Z. and Li, P., Normal families of meromorphic mappings of several complex variables into PN(C) for moving targets, Sci. China Ser. A 48 (2005), suppl., 355-364. MR 2156516. DOI 10.1007/BF02884720.Google Scholar
[23] Zalcman, L., Normal families: New perspectives Bull. Amer. Math. Soc. (N.S.) 35 (1998), 215–230. MR 1624862. DOI 10.1090/S0273-0979-98-00755-1.Google Scholar