Deformation limit and bimeromorphic embedding of Moishezon manifolds

Let φ: → Δ be a holomorphic family of compact complex manifolds over an open disk in C. If the fiber φ-1(t) for each nonzero t in an uncountable subset B of Δ is Moishezon and the reference fiber X0 satisfies the local deformation invariance for Hodge number of type (0, 1) or admits a strongly Gauduchon metric introduced by D. Popovici, then X0 is still Moishezon. We also obtain a bimeromorphic embedding-→ PN × Δ. 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 φ: → Δ. S.-T. Yau's solutions to certain degenerate Monge-Amp?re equations are used. ? 2020 World Scientific Publishing Company.