Proper holomorphic mappings between invariant domains in $\mathbb {C}^n$
HTML articles powered by AMS MathViewer
- by Jiafu Ning, Huiping Zhang and Xiangyu Zhou PDF
- Trans. Amer. Math. Soc. 369 (2017), 517-536 Request permission
Abstract:
In the present paper, we prove the following result generalizing some well-known related results about biholomorphic or proper holomorphic mappings between some special domains in $\mathbb {C}^n$. Let $G_1$ and $G_2$ be two compact Lie groups, which act linearly on $\mathbb {C}^n$ with $\mathcal {O}(\mathbb {C}^n)^{G_j}=\mathbb {C}$ for $j=1,2$. Let $0\in \Omega _j$ be bounded $G_j$-invariant domains in $\mathbb {C}^n$ for $j=1,2$. If $f:\Omega _1\rightarrow \Omega _2$ is a proper holomorphic mapping, then $f$ extends holomorphically to an open neighborhood of $\overline {\Omega }_1$, and in addition if $f^{-1}(0)=\{0\}$, then $f$ is a polynomial mapping. We also prove that if $0\in \Omega$ is a $G_1$-invariant pseudoconvex domain in $\mathbb {C}^n$ with $\mathcal {O}(\mathbb {C}^n)^{G_1}=\mathbb {C}$, then $\Omega$ is orbit convex. The second result is used to prove the first one.References
- David E. Barrett, Regularity of the Bergman projection on domains with transverse symmetries, Math. Ann. 258 (1981/82), no. 4, 441–446. MR 650948, DOI 10.1007/BF01453977
- Eric Bedford, Proper holomorphic mappings, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 157–175. MR 733691, DOI 10.1090/S0273-0979-1984-15235-2
- E. Bedford and S. Bell, Boundary behavior of proper holomorphic correspondences, Math. Ann. 272 (1985), no. 4, 505–518. MR 807287, DOI 10.1007/BF01455863
- Steven R. Bell, Proper holomorphic mappings and the Bergman projection, Duke Math. J. 48 (1981), no. 1, 167–175. MR 610182
- Steven R. Bell, The Bergman kernel function and proper holomorphic mappings, Trans. Amer. Math. Soc. 270 (1982), no. 2, 685–691. MR 645338, DOI 10.1090/S0002-9947-1982-0645338-1
- Steven R. Bell, Proper holomorphic mappings between circular domains, Comment. Math. Helv. 57 (1982), no. 4, 532–538. MR 694605, DOI 10.1007/BF02565875
- Steven Bell, Proper holomorphic mappings that must be rational, Trans. Amer. Math. Soc. 284 (1984), no. 1, 425–429. MR 742433, DOI 10.1090/S0002-9947-1984-0742433-5
- Steve Bell, Proper holomorphic correspondences between circular domains, Math. Ann. 270 (1985), no. 3, 393–400. MR 774364, DOI 10.1007/BF01473434
- Steven Bell and David Catlin, Boundary regularity of proper holomorphic mappings, Duke Math. J. 49 (1982), no. 2, 385–396. MR 659947
- Steven R. Bell and Raghavan Narasimhan, Proper holomorphic mappings of complex spaces, Several complex variables, VI, Encyclopaedia Math. Sci., vol. 69, Springer, Berlin, 1990, pp. 1–38. MR 1095089
- H. Cartan, Les fonctions de deux variables complexes et le problème de représentation analytique, J. de Math. Pures et Appl. 96 (1931), 1-114.
- So-Chin Chen, Regularity of the Bergman projection on domains with partial transverse symmetries, Math. Ann. 277 (1987), no. 1, 135–140. MR 884651, DOI 10.1007/BF01457283
- So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297, DOI 10.1090/amsip/019
- Fusheng Deng, Huiping Zhang, and Xiangyu Zhou, Positivity of direct images of positively curved volume forms, Math. Z. 278 (2014), no. 1-2, 347–362. MR 3267582, DOI 10.1007/s00209-014-1318-2
- Fusheng Deng and Xiangyu Zhou, Rigidity of automorphism groups of invariant domains in certain Stein homogeneous manifolds, C. R. Math. Acad. Sci. Paris 350 (2012), no. 7-8, 417–420 (English, with English and French summaries). MR 2922096, DOI 10.1016/j.crma.2012.02.009
- F.-Sh. Den and Shch.-Yuĭ Chzhou, Rigidity of automorphism groups for invariant domains in homogeneous Stein spaces, Izv. Ross. Akad. Nauk Ser. Mat. 78 (2014), no. 1, 37–64 (Russian, with Russian summary); English transl., Izv. Math. 78 (2014), no. 1, 34–58. MR 3204658, DOI 10.1070/im2014v078n01abeh002679
- Klas Diederich and John Erik Fornaess, Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions, Invent. Math. 39 (1977), no. 2, 129–141. MR 437806, DOI 10.1007/BF01390105
- Klas Diederich and John Erik Fornæss, Boundary regularity of proper holomorphic mappings, Invent. Math. 67 (1982), no. 3, 363–384. MR 664111, DOI 10.1007/BF01398927
- Klas Diederich and John E. Fornæss, Proper holomorphic images of strictly pseudoconvex domains, Math. Ann. 259 (1982), no. 2, 279–286. MR 656667, DOI 10.1007/BF01457314
- Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65. MR 350069, DOI 10.1007/BF01406845
- Franc Forstnerič, Proper holomorphic mappings: a survey, Several complex variables (Stockholm, 1987/1988) Math. Notes, vol. 38, Princeton Univ. Press, Princeton, NJ, 1993, pp. 297–363. MR 1207867
- Peter Heinzner, On the automorphisms of special domains in $\mathbf C^n$, Indiana Univ. Math. J. 41 (1992), no. 3, 707–712. MR 1189907, DOI 10.1512/iumj.1992.41.41037
- Peter Heinzner, Geometric invariant theory on Stein spaces, Math. Ann. 289 (1991), no. 4, 631–662. MR 1103041, DOI 10.1007/BF01446594
- Peter Heinzner and Alan Huckleberry, Invariant plurisubharmonic exhaustions and retractions, Manuscripta Math. 83 (1994), no. 1, 19–29. MR 1265915, DOI 10.1007/BF02567597
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- Wilhelm Kaup, Über das Randverhalten von holomorphen Automorphismen beschränkter Gebiete, Manuscripta Math. 3 (1970), 257–270 (German, with English summary). MR 277769, DOI 10.1007/BF01338659
- Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625, DOI 10.1090/chel/340
- F. Rong, On automorphisms of quasi-circular domains fixing the origin, preprint (arXiv:1403.7769).
- A. G. Sergeev and Shchan′yuĭ Chzhou, On invariant domains of holomorphy, Trudy Mat. Inst. Steklov. 203 (1994), no. Izbran. Voprosy Mat. Fiz. i Anal., 159–172 (Russian); English transl., Proc. Steklov Inst. Math. 3(203) (1995), 145–155. MR 1382601
- Dennis M. Snow, Reductive group actions on Stein spaces, Math. Ann. 259 (1982), no. 1, 79–97. MR 656653, DOI 10.1007/BF01456830
- Karl Stein, Topics on holomorphic correspondences, Rocky Mountain J. Math. 2 (1972), no. 3, 443–463. MR 311945, DOI 10.1216/RMJ-1972-2-3-443
- R. F. Streater and A. S. Wightman, PCT, spin and statistics, and all that, Princeton Landmarks in Physics, Princeton University Press, Princeton, NJ, 2000. Corrected third printing of the 1978 edition. MR 1884336
- XiaoWen Wu, FuSheng Deng, and XiangYu Zhou, Rigidity and regularity in group actions, Sci. China Ser. A 51 (2008), no. 4, 819–826. MR 2395425, DOI 10.1007/s11425-008-0062-7
- Atsushi Yamamori, Automorphisms of normal quasi-circular domains, Bull. Sci. Math. 138 (2014), no. 3, 406–415. MR 3206476, DOI 10.1016/j.bulsci.2013.10.002
- A. Yamamori, The linearity of origin-preserving automorphisms of quasi-circular domains, preprint (arXiv:1404.0309v1).
- Shchan′yuĭ Chzhou, On the orbital convexity of domains of holomorphy that are invariant with respect to the linear action of tori, Dokl. Akad. Nauk SSSR 322 (1992), no. 2, 262–267 (Russian); English transl., Soviet Math. Dokl. 45 (1992), no. 1, 93–98. MR 1158962
- Xiang Yu Zhou, On orbit connectedness, orbit convexity, and envelopes of holomorphy, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 2, 196–205; English transl., Russian Acad. Sci. Izv. Math. 44 (1995), no. 2, 403–413. MR 1275909, DOI 10.1070/IM1995v044n02ABEH001604
- Xiang-Yu Zhou, On invariant domains in certain complex homogeneous spaces, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 4, 1101–1115 (English, with English and French summaries). MR 1488246
- Xiangyu Zhou, Some results related to group actions in several complex variables, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 743–753. MR 1957081
Additional Information
- Jiafu Ning
- Affiliation: College of Mathematics and Statistics, Chongqing University, Chongqing 401331, People’s Republic of China
- Email: jfning@cqu.edu.cn
- Huiping Zhang
- Affiliation: Department of Mathematics, Information School, Renmin University of China, Beijing 100872, People’s Republic of China
- MR Author ID: 717149
- Email: huipingzhang@ruc.edu.cn
- Xiangyu Zhou
- Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Science, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
- MR Author ID: 260186
- Email: xyzhou@math.ac.cn
- Received by editor(s): December 18, 2013
- Received by editor(s) in revised form: August 19, 2014, and January 9, 2015
- Published electronically: May 6, 2016
- Additional Notes: The authors were partially supported by NSFC. The first author was supported by the Fundamental Research Funds for the Central Universities (Project No.0208005202035)
The second author is the corresponding author - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 517-536
- MSC (2010): Primary 32D05, 32H35, 32H40, 32M05, 32T05
- DOI: https://doi.org/10.1090/tran/6690
- MathSciNet review: 3557783