Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5778460 | Advances in Mathematics | 2017 | 38 Pages |
Abstract
We introduce the Hermitian-invariant group Îf 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 Îf is non-compact if and only if f is a totally geodesic embedding. We show that Îf contains an n-torus if and only if f is equivalent to a monomial map. We show that Îf 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 Îf (when the target is a generalized ball) to contain automorphisms that move the origin.
Keywords
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Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
John P. D'Angelo, Ming Xiao,