Topological Aspects of Holomorphic Mappings of Hyperquadrics from $\mathbb C^2$ to $\mathbb C^3$

TL;DR

This paper investigates the topological properties of holomorphic mappings from $ ext{C}^2$ to $ ext{C}^3$ that map hyperquadrics, revealing significant differences based on the target's signature and establishing structural results for certain subclasses.

Contribution

It extends previous results to analyze the topological structure of nondegenerate holomorphic mappings between hyperquadrics, highlighting differences based on target signatures and exploring automorphism actions.

Findings

Discreteness of the class modulo automorphisms when the target is a sphere.

Failure of discreteness when the target hyperquadric has signature (2,1).

Structural insights into automorphism actions on specific subsets of mappings.

Abstract

Based on the results in [Rei14a] we deduce some topological results concerning holomorphic mappings of Levi-nondegenerate hyperquadrics under biholomorphic equivalence. We study the class $\mathcal F$ of so-called nondegenerate and transversal holomorphic mappings sending locally 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 of automorphisms fixing a given point on $\mathcal F$ to obtain a structural result for a particularly interesting subset of $\mathcal F$.