A remark on the embedding theorem associated to complex connections of mixed type

Abstract

Let $M$ be a compact complex manifold and let $(L,H)$ be a holomorphic Hermitian line bundle over $M$ such that the curvature form of $h$ is nondegenerate and splits into the difference $\Theta_{+} - \Theta_{-}$ of two semipositive forms $\Theta_{+}$ and $\Theta_{-}$ whose null spaces define mutually transverse holomorphic foliations $\mathcal{F}_{-}$ and $\mathcal{F}_{+}$, respectively. Then $L^{m}$ admits, for sufficiently large $m \in \mathbb{N}$, $C^{\infty}$ sections whose ratio embeds $M$ into $\mathbb{CP}^{N}$ holomorphically (resp. antiholomorphically) along $\mathcal{F}_{+}$ (resp. along $\mathcal{F}_{-}$).