Topological aspects of holomorphic mappings of hyperquadrics from $\mathbb C^2$ to $\mathbb C^3$

Michael Reiter

Vol. 280 (2016), No. 2, 455–474

DOI: 10.2140/pjm.2016.280.455

Abstract

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 ${ℂ}^2$ to a Levi-nondegenerate hyperquadric in ${ℂ}^3$, 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 $\left(2,1\right)$. 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 $\mathcal{ℱ}$.

Keywords

holomorphic mappings, hyperquadrics, nondegeneracy, transversality, automorphism groups, group actions, quotient space, principal fiber bundle

Mathematical Subject Classification 2010

Primary: 32H02, 32V30, 57S05, 57S25, 58D19

Milestones

Received: 6 November 2014

Revised: 20 May 2015

Accepted: 8 July 2015

Published: 28 January 2016