Characterize holomorphic retracts of C^2

Characterize all holomorphic retracts of the complex Euclidean plane C^2 by determining a complete description (up to automorphisms of C^2) of the closed complex submanifolds that arise as images of holomorphic retraction maps on C^2.

Background

The paper develops systematic results on holomorphic retracts of various classes of domains, including bounded balanced pseudoconvex domains, polyballs, the Hartogs triangle, and certain unbounded products. For C^2 specifically, the authors point out that while many retracts are graphs (e.g., graphs of entire functions) and while retracts of products of planar domains often appear as graphs over a factor, not every retract of C^2 has this form.

They provide explicit examples demonstrating that some holomorphic retracts of C^2 are not graphs over either coordinate factor and recall that all polynomial retracts of C^2 are classifiable (Shpilrain–Yu) and rectifiable to a coordinate axis by a polynomial automorphism. Moreover, by adapting a construction of Forstnerič–Globevnik–Rosay, they exhibit non-straightenable retracts, highlighting the diversity of holomorphic retracts beyond the polynomial category.

Against this backdrop, the authors note that a full classification of holomorphic retracts of C^2 remains to be achieved, motivating an open problem to determine necessary and sufficient conditions (or an explicit description) for all such retracts.