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Hermitian Symmetric Polynomials and CR Complexity

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Abstract

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.

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References

  1. Baouendi, M.S., Ebenfelt, P., Huang, X.: Holomorphic mappings between hyperquadrics with small signature difference. arXiv:0906.1235

  2. Baouendi, M.S., Huang, X.: Super-rigidity for holomorphic mappings between hyperquadrics with positive signature. J. Differ. Geom. 69(2), 379–398 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Catlin, D., D’Angelo, J.: A stabilization theorem for Hermitian forms and applications to holomorphic mappings. Math. Res. Lett. 3, 149–166 (1996)

    MathSciNet  MATH  Google Scholar 

  4. D’Angelo, J.: Several Complex Variables and the Geometry of Real Hypersurfaces. CRC Press, Boca Raton (1993)

    MATH  Google Scholar 

  5. D’Angelo, J.: Inequalities from Complex Analysis. Carus Mathematical Monograph, No. 28. Mathematics Association of America, Washington (2002)

    MATH  Google Scholar 

  6. D’Angelo, J.: Proper holomorphic mappings, positivity conditions, and isometric imbedding. J. Korean Math. Soc. 1–30 (2003)

  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. D’Angelo, J., Lebl, J.: Complexity results for CR mappings between spheres. Int. J. Math. 20(2), 149–166 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  11. D’Angelo, J., Lebl, J., Peters, H.: Degree estimates for polynomials constant on hyperplanes. Mich. Math. J. 55(3), 693–713 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ebenfelt, P., Huang, X., Zaitsev, D.: Rigidity of CR-immersions into spheres. Commun. Anal. Geom. 12(3), 631–670 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Faran, J.: Maps from the two-ball to the three-ball. Invent. Math. 68, 441–475 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Faran, J.: On the linearity of proper maps between balls in the low codimension case. J. Differ. Geom. 24, 15–17 (1986)

    MathSciNet  MATH  Google Scholar 

  15. Forstneric, F.: Extending proper holomorphic maps of positive codimension. Invent. Math. 95, 31–62 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  17. Huang, X., Ji, S.: Mapping B n into B 2n−1. Invent. Math. 145, 219–250 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  19. Lebl, J.: Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics. arXiv: 0906.0325

  20. Lebl, J., Peters, H.: Polynomials constant on a hyperplane and CR maps of hyperquadrics. arXiv: 0910.2673

  21. Marques de Sá, E.: On the inertia of sums of Hermitian matrices. Linear Algebra Appl. 37, 143–159 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana, IL, 61801, USA

    John P. D’Angelo & Jiří Lebl

Authors
  1. John P. D’Angelo
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  2. Jiří Lebl
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Correspondence to John P. D’Angelo.

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Communicated by Alexander Isaev.

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D’Angelo, J.P., Lebl, J. Hermitian Symmetric Polynomials and CR Complexity. J Geom Anal 21, 599–619 (2011). https://doi.org/10.1007/s12220-010-9160-1

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  • DOI: https://doi.org/10.1007/s12220-010-9160-1

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