## Proper holomorphic polynomial maps between bounded symmetric domains of classical type

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- by Aeryeong Seo PDF
- Proc. Amer. Math. Soc.
**144**(2016), 739-751 Request permission

## Abstract:

We prove that two proper holomorphic polynomial maps between bounded symmetric domains of classical type which preserve the origin are equivalent if and only if they are isotropically equivalent. Using this property we show that each member of a one-parameter family of maps from a 2015 paper by the author is inequivalent.## References

- H. Alexander,
*Holomorphic mappings from the ball and polydisc*, Math. Ann.**209**(1974), 249–256. MR**352531**, DOI 10.1007/BF01351851 - John P. D’Angelo,
*Polynomial proper maps between balls*, Duke Math. J.**57**(1988), no. 1, 211–219. MR**952233**, DOI 10.1215/S0012-7094-88-05710-9 - John P. D’Angelo,
*Proper holomorphic maps between balls of different dimensions*, Michigan Math. J.**35**(1988), no. 1, 83–90. MR**931941**, DOI 10.1307/mmj/1029003683 - John P. D’Angelo,
*Polynomial proper holomorphic mappings between balls. II*, Michigan Math. J.**38**(1991), no. 1, 53–65. MR**1091509**, DOI 10.1307/mmj/1029004261 - John P. D’Angelo and Jiří Lebl,
*On the complexity of proper holomorphic mappings between balls*, Complex Var. Elliptic Equ.**54**(2009), no. 3-4, 187–204. MR**2513534**, DOI 10.1080/17476930902759403 - D’Angelo, John P.; Lebl, Jir̆í,
*Homotopy equivalence for proper holomorphic mappings*, arXiv:1408.1104 (2014) - John P. D’Angelo, Jiří Lebl, and Han Peters,
*Degree estimates for polynomials constant on a hyperplane*, Michigan Math. J.**55**(2007), no. 3, 693–713. MR**2372622**, DOI 10.1307/mmj/1197056463 - James J. Faran,
*Maps from the two-ball to the three-ball*, Invent. Math.**68**(1982), no. 3, 441–475. MR**669425**, DOI 10.1007/BF01389412 - James J. Faran,
*The linearity of proper holomorphic maps between balls in the low codimension case*, J. Differential Geom.**24**(1986), no. 1, 15–17. MR**857373** - Franc Forstnerič,
*Extending proper holomorphic mappings of positive codimension*, Invent. Math.**95**(1989), no. 1, 31–61. MR**969413**, DOI 10.1007/BF01394144 - Hidetaka Hamada,
*Rational proper holomorphic maps from $\mathbf B^n$ into $\mathbf B^{2n}$*, Math. Ann.**331**(2005), no. 3, 693–711. MR**2122546**, DOI 10.1007/s00208-004-0606-2 - Xiaojun Huang,
*On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions*, J. Differential Geom.**51**(1999), no. 1, 13–33. MR**1703603** - Xiaojun Huang and Shanyu Ji,
*Mapping $\mathbf B^n$ into $\mathbf B^{2n-1}$*, Invent. Math.**145**(2001), no. 2, 219–250. MR**1872546**, DOI 10.1007/s002220100140 - Xiaojun Huang, Shanyu Ji, and Wanke Yin,
*On the third gap for proper holomorphic maps between balls*, Math. Ann.**358**(2014), no. 1-2, 115–142. MR**3157993**, DOI 10.1007/s00208-013-0952-z - Xiaojun Huang, Shanyu Ji, and Dekang Xu,
*A new gap phenomenon for proper holomorphic mappings from $B^n$ into $B^N$*, Math. Res. Lett.**13**(2006), no. 4, 515–529. MR**2250487**, DOI 10.4310/MRL.2006.v13.n4.a2 - Kim, Sung-Yeon; Zaitsev, Dmitri,
*Rigidity of proper holomorphic maps between bounded symmetric domains*, to appear Math. Ann. in 2015. - Jiří Lebl and Han Peters,
*Polynomials constant on a hyperplane and CR maps of spheres*, Illinois J. Math.**56**(2012), no. 1, 155–175 (2013). MR**3117023** - Ottmar Loos,
*Jordan pairs*, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR**0444721** - Ng, Sui-Chung,
*Holomorphic double fibration and the mapping problems of classical domains*, Int Math Res Notices first published online September 18, 2013 doi:10.1093/imrn/rnt200. - Sui-Chung Ng,
*On proper holomorphic mappings among irreducible bounded symmetric domains of rank at least 2*, Proc. Amer. Math. Soc.**143**(2015), no. 1, 219–225. MR**3272747**, DOI 10.1090/S0002-9939-2014-12226-X - Seo, Aeryeong,
*New examples of proper holomorphic maps among symmetric domains*, to be appeared in Michigan Math. J on 2015. - I Hsun Tsai,
*Rigidity of proper holomorphic maps between symmetric domains*, J. Differential Geom.**37**(1993), no. 1, 123–160. MR**1198602** - Zhen-Han Tu,
*Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains*, Math. Z.**240**(2002), no. 1, 13–35. MR**1906705**, DOI 10.1007/s002090100353 - Zhen-Han Tu,
*Rigidity of proper holomorphic mappings between equidimensional bounded symmetric domains*, Proc. Amer. Math. Soc.**130**(2002), no. 4, 1035–1042. MR**1873777**, DOI 10.1090/S0002-9939-01-06383-3 - S. M. Webster,
*The rigidity of C-R hypersurfaces in a sphere*, Indiana Univ. Math. J.**28**(1979), no. 3, 405–416. MR**529673**, DOI 10.1512/iumj.1979.28.28027 - Joseph A. Wolf,
*Fine structure of Hermitian symmetric spaces*, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Pure and App. Math., Vol. 8, Dekker, New York, 1972, pp. 271–357. MR**0404716**

## Additional Information

**Aeryeong Seo**- Affiliation: School of Mathematics, Korea Institute for Advanced Study (KIAS), 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Korea
- MR Author ID: 984919
- Email: Aileen83@kias.re.kr
- Received by editor(s): November 27, 2014
- Received by editor(s) in revised form: December 24, 2014, January 16, 2015, and February 3, 2015
- Published electronically: June 30, 2015
- Communicated by: Franc Forstneric
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 739-751 - MSC (2010): Primary 32M15, 32H35
- DOI: https://doi.org/10.1090/proc12755
- MathSciNet review: 3430850