Hermitian Symmetric Polynomials and CR Complexity
- 62 Downloads
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.
KeywordsHermitian 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.
- 1.Baouendi, M.S., Ebenfelt, P., Huang, X.: Holomorphic mappings between hyperquadrics with small signature difference. arXiv:0906.1235
- 6.D’Angelo, J.: Proper holomorphic mappings, positivity conditions, and isometric imbedding. J. Korean Math. Soc. 1–30 (2003) Google Scholar
- 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
- 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