Homogenization, Reflection, and the X-variety

John P. D'Angelo

Research output: Contribution to journalArticlepeer-review


The author uses a homogenization technique to analyze a reflection principle in complex analysis. Given a holomorphic polynomial on complex Euclidean space, the author introduces its reflection, which is a matrix of anti-holomorphic polynomials, and establishes basic properties. The introduction to the paper discusses the classical one variable case, where the reflection is well known. The main result shows that the X-variety, defined by Forstneric in his study of proper mappings between balls, is an affine space determined by the null space of the reflection matrix. An integral formula for this matrix is also given. The author provides examples both when the X-variety is a bundle and when it is not a bundle, and then proves several general results to this effect. The final result gives a new family of examples where the X-variety of a map equals the graph of the map.

Original languageEnglish (US)
Pages (from-to)1113-1133
Number of pages21
JournalIndiana University Mathematics Journal
Issue number5
StatePublished - 2003


  • Proper holomorphic mappings
  • Reflection principle
  • Several complex variables
  • Unit ball

ASJC Scopus subject areas

  • General Mathematics


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