Communications in Contemporary Mathematics ( IF 1.278 ) Pub Date : 2020-11-30 , DOI: 10.1142/s021919972050087x
Sheng Rao; I-Hsun Tsai

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.

Moishezon流形的变形极限和双态嵌入

$π：𝒳→Δ$ 是在开放磁盘上的紧凑型复杂流形的全同族 $ℂ$。如果纤维$π-1个（Ť）$ 对于每个非零 $Ť$ 在不可数的子集中 $乙$$Δ$ 是Moishezon和参考纤维 $X0$ 满足Hodge型数的局部变形不变性 $（0，1个）$ 或接受D. Popovici引入的强Gauduchon度量，然后 $X0$仍然是Moishezon。我们还获得了双态嵌入$𝒳-→ℙñ×Δ$。我们的证明可以看作是Popovici在2009年，2010年和2013年提出并证明的与此方向上的若干结果的新的代数证明。$0$ 不一定是 $乙$和双态嵌入是新的。我们的证明策略在于，在全纯族的整个空间上构造一个全局全纯线束，并研究双全形的几何$π：𝒳→Δ$。S.-T. 使用Yau对某些退化的Monge-Ampère方程的解。

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