Symmetries in CR complexity theory

John P. D'Angelo, Ming Xiao

Research output: Contribution to journalArticlepeer-review


We introduce the Hermitian-invariant group Γ f of a proper rational map f between the unit ball in complex Euclidean space and a generalized ball in a space of typically higher dimension. We use properties of the groups to define the crucial new concepts of essential map and the source rank of a map. We prove that every finite subgroup of the source automorphism group is the Hermitian-invariant group of some rational proper map between balls. We prove that Γ f is non-compact if and only if f is a totally geodesic embedding. We show that Γ f contains an n-torus if and only if f is equivalent to a monomial map. We show that Γ f contains a maximal compact subgroup if and only if f is equivalent to the juxtaposition of tensor powers. We also establish a monotonicity result; the group, after intersecting with the unitary group, does not decrease when a tensor product operation is applied to a polynomial proper map. We give a necessary condition for Γ f (when the target is a generalized ball) to contain automorphisms that move the origin.

Original languageEnglish (US)
Pages (from-to)590-627
Number of pages38
JournalAdvances in Mathematics
StatePublished - Jun 20 2017


  • Automorphism groups
  • CR complexity
  • Group-invariant CR maps
  • Hermitian forms
  • Proper holomorphic mappings
  • Unitary transformations

ASJC Scopus subject areas

  • Mathematics(all)


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