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Topological aspects of holomorphic mappings of hyperquadrics from \(\mathbb{C}^2\) to \(\mathbb{C}^3\). (English) Zbl 1332.32019

Summary: In this article we deduce some topological results concerning holomorphic mappings of hyperquadrics under biholomorphic equivalence. We study the class \(\mathcal F\) of so-called nondegenerate and transversal holomorphic mappings locally sending the sphere in \(\mathbb C^2\) to a Levi-nondegenerate hyperquadric in \(\mathbb C^3\), which contains the most interesting mappings. We show that from a topological point of view there is a major difference when the target is the sphere or the hyperquadric with signature \((2,1)\). In the first case, \(\mathcal F\) modulo the group of automorphisms is discrete, in contrast to the second case, where this property fails to hold. Furthermore, we study some basic properties such as freeness and properness of the action on \(\mathcal F\) of automorphisms fixing a given point to obtain a structural result for a particularly interesting subset of \(\mathcal F\).

MSC:

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32V30 Embeddings of CR manifolds
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
57S25 Groups acting on specific manifolds
58D19 Group actions and symmetry properties
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