$1$-complete semiholomorphic foliations

Abstract:

A semiholomorphic foliation 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 $\mathbb {C}^N$. 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 $\mathbb {CP}^N$.