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 and `analytic complements' thereof. Thereafter, similar characterization results for domains which are neither bounded nor topologically trivial. We conclude by reporting some results on the retracts of \(\mathbb{C}^2\).