Journal of Geometric Analysis

, Volume 21, Issue 3, pp 599–619 | Cite as

Hermitian Symmetric Polynomials and CR Complexity



Properties of Hermitian forms are used to investigate several natural questions from CR geometry. To each Hermitian symmetric polynomial we assign a Hermitian form. We study how the signature pairs of two Hermitian forms behave under the polynomial product. We show, except for three trivial cases, that every signature pair can be obtained from the product of two indefinite forms. We provide several new applications to the complexity theory of rational mappings between hyperquadrics, including a stability result about the existence of non-trivial rational mappings from a sphere to a hyperquadric with a given signature pair.


Hermitian forms Embeddings of CR manifolds Hyperquadrics Signature pairs CR complexity theory Proper holomorphic mappings 

Mathematics Subject Classification (2000)

15B57 32V30 32H35 14P05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baouendi, M.S., Ebenfelt, P., Huang, X.: Holomorphic mappings between hyperquadrics with small signature difference. arXiv:0906.1235
  2. 2.
    Baouendi, M.S., Huang, X.: Super-rigidity for holomorphic mappings between hyperquadrics with positive signature. J. Differ. Geom. 69(2), 379–398 (2005) MathSciNetMATHGoogle Scholar
  3. 3.
    Catlin, D., D’Angelo, J.: A stabilization theorem for Hermitian forms and applications to holomorphic mappings. Math. Res. Lett. 3, 149–166 (1996) MathSciNetMATHGoogle Scholar
  4. 4.
    D’Angelo, J.: Several Complex Variables and the Geometry of Real Hypersurfaces. CRC Press, Boca Raton (1993) MATHGoogle Scholar
  5. 5.
    D’Angelo, J.: Inequalities from Complex Analysis. Carus Mathematical Monograph, No. 28. Mathematics Association of America, Washington (2002) MATHGoogle Scholar
  6. 6.
    D’Angelo, J.: Proper holomorphic mappings, positivity conditions, and isometric imbedding. J. Korean Math. Soc. 1–30 (2003) Google Scholar
  7. 7.
    D’Angelo, J.: Invariant CR mappings. In: Complex Analysis: Several Complex Variables and Connections with PDEs and Geometry (Fribourg 2008). Trends in Mathematics, pp. 95–107. Birkhäuser, Basel (2010) Google Scholar
  8. 8.
    D’Angelo, J., Lebl, J.: Complexity results for CR mappings between spheres. Int. J. Math. 20(2), 149–166 (2009) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    D’Angelo, J., Lebl, J.: On the complexity of proper holomorphic mappings between balls. Complex Var. Elliptic Equ. 54(3–4), 187–204 (2009) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    D’Angelo, J., Kos, Š., Riehl, E.: A sharp bound for the degree of proper monomial mappings between balls. J. Geom. Anal. 13(4), 581–593 (2003) MathSciNetMATHGoogle Scholar
  11. 11.
    D’Angelo, J., Lebl, J., Peters, H.: Degree estimates for polynomials constant on hyperplanes. Mich. Math. J. 55(3), 693–713 (2007) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Ebenfelt, P., Huang, X., Zaitsev, D.: Rigidity of CR-immersions into spheres. Commun. Anal. Geom. 12(3), 631–670 (2004) MathSciNetMATHGoogle Scholar
  13. 13.
    Faran, J.: Maps from the two-ball to the three-ball. Invent. Math. 68, 441–475 (1982) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Faran, J.: On the linearity of proper maps between balls in the low codimension case. J. Differ. Geom. 24, 15–17 (1986) MathSciNetMATHGoogle Scholar
  15. 15.
    Forstneric, F.: Extending proper holomorphic maps of positive codimension. Invent. Math. 95, 31–62 (1989) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Huang, X.: On a linearity problem for proper maps between balls in complex spaces of different dimensions. J. Differ. Geom. 51(1), 13–33 (1999) MATHGoogle Scholar
  17. 17.
    Huang, X., Ji, S.: Mapping B n into B 2n−1. Invent. Math. 145, 219–250 (2001) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Huang, X., Ji, S., Xu, D.: A new gap phenomenon for proper holomorphic mappings from B n into B N. Math. Res. Lett. 13(4), 515–529 (2006) MathSciNetMATHGoogle Scholar
  19. 19.
    Lebl, J.: Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics. arXiv: 0906.0325
  20. 20.
    Lebl, J., Peters, H.: Polynomials constant on a hyperplane and CR maps of hyperquadrics. arXiv: 0910.2673
  21. 21.
    Marques de Sá, E.: On the inertia of sums of Hermitian matrices. Linear Algebra Appl. 37, 143–159 (1981) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Meylan, F.: Degree of a holomorphic map between unit balls from C 2 to C n. Proc. Am. Math. Soc. 134(4), 1023–1030 (2006) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Dept. of MathematicsUniv. of IllinoisUrbanaUSA

Personalised recommendations