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Limits of Moishezon Manifolds under Holomorphic Deformations

Given a (smooth) complex analytic family of compact complex manifolds, we prove that the central fibre must be Moishezon if the other fibres are Moishezon. Using a "strongly Gauduchon metric" on the central fibre whose existence was proved in our previous work on limits of projective manifolds, we show that the irreducible components of the relative Barlet space of divisors contained in the fibres are proper over the base even under the weaker assumption that the $\partial\bar\partial$-lemma hold on all the fibres except, possibly, the central one. This implies that the algebraic dimension of the central fibre cannot be lower than that of the generic fibre. Since the latter is already maximal thanks to the Moishezon assumption, the central fibre must be of maximal algebraic dimension, hence Moishezon.