A generalization of Matsushima’s embedding theorem

Abstract

Let $M$ be a compact complex manifold and let $L \rightarrow M$ be a homorphic line bundle whose curvature form is everywhere of signature $(s_+, s_-)$. Under some conditions on the curvature form of $L$, it is show that $K \otimes L^{\otimes m}$ admits, for some $K$ and sufficiently large $m \in \mathbb{N}$, $C^{\infty}$ sections $t_0, \ldots, t_N$ such that the ratio $(t_0 : \cdots : t_N)$ embeds $M$ holomorphically in $s_+$ variables and antiholomorphically in $s_-$ variables. The result extends the Kodaira's embedding theorem as well as aresult of Matsushima for complex tori.