The generalized Oka-Grauert principle for 1-convex manifolds

TL;DR

This paper proves a generalized Oka-Grauert principle for 1-convex manifolds, showing that certain continuous maps can be homotoped to holomorphic maps under specific conditions, expanding the understanding of complex structures.

Contribution

It extends the Oka-Grauert principle to 1-convex manifolds, allowing homotopies to holomorphic maps when Y satisfies CAP or the complex structure on X can be altered.

Findings

Homotopic to holomorphic maps under specified conditions

Extension of Oka-Grauert principle to 1-convex manifolds

Conditions involving CAP or adjustable complex structures

Abstract

This paper presents a proof of the generalized Oka-Grauert principle for 1-convex manifolds: Every continuous mapping from a 1-convex manifold X to a complex manifold Y which is already holomorphic on a neighborhood of the exceptional set is homotopic to a holomorphic one provided that either Y satisfies CAP or we are free to change the complex structure on X.