Deformation limit and bimeromorphic embedding of Moishezon manifolds

Let $\pi :\mathcal{X}\to \mathrm{\Delta }$ be a holomorphic family of compact complex manifolds over an open disk in $\mathbb{C}$. If the fiber ${\pi }^{-1}(t)$ for each nonzero $t$ in an uncountable subset $B$ of $\mathrm{\Delta }$ 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 $\mathcal{X}⇢{\mathbb{P}}^N\times \mathrm{\Delta }$. 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 $\pi :\mathcal{X}\to \mathrm{\Delta }$. S.-T. Yau’s solutions to certain degenerate Monge–Amp`ere equations are used.