Oka principle for Levi flat manifolds

TL;DR

This paper explores the extension of the Oka principle to Levi-flat manifolds, focusing on classifying CR-bundles on certain semiholomorphic foliations, marking initial steps in this research area.

Contribution

It initiates the study of the Oka principle on Levi-flat manifolds, specifically addressing the classification of CR-bundles in particular cases.

Findings

Partial classification results for CR-bundles on semiholomorphic foliations.

Identification of specific cases where the Oka principle may hold on Levi-flat manifolds.

Preliminary insights into the holomorphic versus continuous problem-solving on Levi-flat structures.

Abstract

The name of Oka principle, or Oka-Grauert principle, is traditionally used to refer to the holomorphic incarnation of the homotopy principle: on a Stein space, every problem that can be solved in the continuous category, can be solved in the holomorphic category as well. In this note, we begin the study of the same kind of questions on a Levi-flat manifold; more precisely, we try to obtain a classification of CR-bundles on a semiholomorphic foliation of type (n, 1). Our investigation should only be considered a preliminary exploration, as it deals only with some particular cases, either in terms of regularity or bidegree of the bundle, and partial results.