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 $B$. As for retracts not passing through origin, we obtain the following result: if $B$ is a strictly convex ball and $\rho$ any holomorphic retraction map on $B$ which is submersive at its center, then $Z=\rho(B)$ is the graph of a holomorphic map over a linear subspace of $B$. To deal with a case when $\partial B$ may fail to have sufficiently many extreme points, we consider products of strictly convex balls, with respect to various norms 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 $M$ which can be expressed as an increasing union of submanifolds which are biholomorphic to a prescribed homogeneous bounded balanced domain. 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 $0<q\<1$; 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 andanalytic complements' thereof. Thereafter, similar characterization results for domains which are neither bounded nor topologically trivial. We conclude with some expositions about retracts of $\mathbb{C}^2$.