Recent developments on Oka manifolds

TL;DR

This paper reviews recent progress in Oka theory, introduces new examples and results on Oka domains, and addresses a question about degeneration of fibers in Stein families, advancing understanding of complex manifold properties.

Contribution

It provides an overview of developments since 2017, introduces new Oka domains, and answers a question about fiber degeneration in Stein families, expanding the theory's scope.

Findings

New examples of Oka domains in Euclidean and projective spaces

Degeneration of $\

Answering Ohsawa's question on fiber degeneration in Stein families.

Abstract

In this paper we present the main developments in Oka theory since the publication of my book Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis)}, Second Edition, Springer, 2017. We also give several new results, examples and constructions of Oka domains in Euclidean and projective spaces. Furthermore, we show that for $n>1$ the fibre $\mathbb C^n$ in a Stein family can degenerate to a non-Oka fibre, thereby answering a question of Takeo Ohsawa. Several open problems are discussed.