Recent developments on Oka manifolds

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.

Dedicated to Jaap Korevaar in honour of his 100th birthday. Research was supported by the European Union (ERC Advanced grant HPDR, 101053085) and grants P1-0291, J1-3005, and N1-0237 from ARRS, Republic of Slovenia

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Cited by in corpus (10)

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• Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
• Universal holomorphic maps with slow growth I. An Algorithm
• Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure
• Algebraic Gromov ellipticity: a brief survey
• Oka-1 manifolds: New examples and properties
• Oka tubes in holomorphic line bundles
• Oka-1 manifolds
• Proper holomorphic embeddings with small limit sets