Deformation limit and bimeromorphic embedding of Moishezon manifolds

Abstract

Let $π: \mathcal{X}\rightarrow Δ$ be a holomorphic family of compact complex manifolds over an open disk in $\mathbb{C}$. If the fiber $π^{-1}(t)$ for each nonzero $t$ in an uncountable subset $B$ of $Δ$ is Moishezon and the reference fiber $X_0$ satisfies the local deformation invariance for Hodge number of type $(0,1)$ or admits a strongly Gauduchon metric introduced by D. Popovici, then $X_0$ is still Moishezon. We also obtain a bimeromorphic embedding $\mathcal{X}\dashrightarrow\mathbb{P}^N\timesΔ$. Our proof can be regarded as a new, algebraic proof of several results in this direction proposed and proved by Popovici in 2009, 2010 and 2013. However, our assumption with $0$ not necessarily being a limit point of $B$ and the bimeromorphic embedding are new. Our strategy of proof lies in constructing a global holomorphic line bundle over the total space of the holomorphic family and studying the bimeromorphic geometry of $π:\mathcal{X}\rightarrow Δ$. S.-T. Yau's solutions to certain degenerate Monge--Ampère equations are used.