Exploring Holomorphic Retracts

Abstract

The purpose of this article is towards systematically characterizing (holomorphic) retracts of domains of holomorphy; to begin with, bounded balanced pseudoconvex domains $B \subset \mathbb{C}^N$. Specifically, we show that every retract of $B$ passing through its center (origin), is the graph of a holomorphic map over a linear subspace of $B$. To deal with a case when may fail to have sufficiently many extreme points, we consider products of bounded balanced domains of holomorphy with holomorphically extreme boundaries and obtain a complete description of retracts passing through its center. This can be applied to solve a special case of the union problem with a degeneracy, namely: to characterize those Kobayashi corank one complex manifolds which can be expressed as an increasing union of submanifolds which are biholomorphic to a prescribed homogeneous bounded balanced domain. In this course, we prove a generalization of Mazet's Schwarz lemma. Results about non-existence of retracts of each possible dimension is established for the simplest non-convex but pseudoconvex domain: the $\ell^q$-`ball' for all $0<q <1$ as well as `anisotropic' analogues. The same is also done for balanced analytic polyhedra. This enables an illustration of applying retracts to establishing biholomorphic inequivalences. To go beyond balanced domains, we then first obtain a complete characterization of retracts of the Hartogs triangle and `analytic complements' thereof. Thereafter, similar characterization results for domains which are neither bounded nor topologically trivial. We conclude with some expositions about retracts of $\mathbb{C}^2$.