Let π : 𝒳 → Δ be a holomorphic family of compact complex manifolds over an open disk in ℂ. If the fiber π−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 𝒳−−→ 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 π : 𝒳 → Δ. S.-T. Yau’s solutions to certain degenerate Monge–Ampère equations are used.

中文翻译：

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Moishezon流形的变形极限和双亚纯嵌入

让π ： 𝒳 → Δ是开盘上紧致复流形的全纯族ℂ. 如果纤维π-1(吨)对于每个非零吨在不可数子集中乙的Δ是 Moishezon 和参考纤维X0满足类型霍奇数的局部变形不变性(0, 1)或承认 D. Popovici 引入的强 Gauduchon 度量，然后X0仍然是Moishezon。我们还获得了双亚态嵌入𝒳——→ ℙñ × Δ. 我们的证明可以看作是波波维奇在 2009 年、2010 年和 2013 年提出并证明的几个结果的新的代数证明。然而，我们的假设与0不一定是极限点乙双亚态嵌入是新的。我们的证明策略在于在全纯族的整个空间上构建一个全局全纯线丛，并研究其双亚纯几何π ： 𝒳 → Δ. 英石。使用了 Yau 对某些退化 Monge-Ampère 方程的解。