Symmetries in CR complexity theory

Abstract

We introduce the Hermitian-invariant group ${\Gamma }_f$ 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 ${\Gamma }_f$ is non-compact if and only if $f$ is a totally geodesic embedding. We show that ${\Gamma }_f$ contains an $n$-torus if and only if $f$ is equivalent to a monomial map. We show that ${\Gamma }_f$ 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 ${\Gamma }_f$ (when the target is a generalized ball) to contain automorphisms that move the origin.

30 pages. To appear in Advances in Mathematics

References in corpus (1)

• On the complexity of proper holomorphic mappings between balls