1-Complete semiholomorphic foliations

TL;DR

This paper introduces a new notion of pseudoconvexity for semiholomorphic foliations, leading to vanishing theorems, extension results, and embedding theorems for these complex-structured manifolds.

Contribution

It defines pseudoconvexity for semiholomorphic foliations and proves key theorems on CR cohomology, extensions, and embeddings, advancing the understanding of these geometric structures.

Findings

Vanishing theorem for CR cohomology of 1-complete foliations

Extension of CR functions on Levi flat hypersurfaces

Embedding theorems in complex projective space

Abstract

A semiholomorphic foliations of type (n, d) is a differentiable real manifold X of dimension 2n + d, foliated by complex leaves of complex dimension n. In the present work, we introduce an appropriate notion of pseudoconvexity (and consequently, q-completeness) for such spaces, given by the interplay of the usual pseudoconvexity, along the leaves, and the positivity of the transversal bundle. For 1-complete real analytic semiholomorphic foliations, we obtain a vanishing theorem for the CR cohomology, which we use to show an extension result for CR functions on Levi flat hypersurfaces and an embedding theorem in C^N . In the compact case, we introduce a notion of weak positivity for the transversal bundle, which allows us to construct a real analytic embedding in CP^N .