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Deformation limit and bimeromorphic embedding of Moishezon manifolds. (English) Zbl 1484.32020

Summary: Let \(\pi:\mathcal{X}\to\Delta\) 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 \(\Delta\) 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 \(\mathcal{X}\dashrightarrow\mathbb{P}^N\times\Delta\). 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 \(\pi:\mathcal{X}\to\Delta\). S.-T. Yau’s solutions to certain degenerate Monge-Ampère equations are used.

MSC:

32G05 Deformations of complex structures
32J18 Compact complex \(n\)-folds
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