Recent developments on Oka manifolds

2023/01/27 by Franc Forstnerič · 6 citations
Mathematics · #Holomorphic and Operator Theory #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows

paper · doi:10.1016/j.indag.2023.01.005

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 ℂn in a Stein family can degenerate to a non-Oka fibre, thereby answering a question of Takeo Ohsawa. Several open problems are discussed.

Cited by

• Oka-1 manifolds: New examples and properties
• Isotopies of complete minimal surfaces of finite total curvature
• The universal family of punctured Riemann surfaces is Stein
• Oka-1 manifolds
• The Oka principle for tame families of Stein manifolds
• From Stein manifolds to Oka manifolds: the h-principle in complex analysis

Related

• Eight lectures on Oka manifolds
• Mapping Cylinders and the Oka Principle
• The Oka principle, lifting of holomorphic maps and removability of intersections
• Deformations of Oka manifolds
• The Oka principle for tame families of Stein manifolds