Upper semicontinuity of algebraic dimension

Abstract

We prove the conjecture that the deformation limit of Moishezon manifolds under a smooth deformation over the unit disc in $\mathbb{C}$ is Moishezon. More generally, for a proper holomorphic submersion over the unit disc with connected compact fibers satisfying the $\partial \bar{\partial }$-lemma except possibly at one parameter, we show that the very general algebraic dimension is the minimum on the entire disc. The proof builds on Barlet's theory of cycle spaces and Rao--Tsai's torsion-freeness method for extending integral Chern classes. Taking this approach as a starting point, we handle the remaining finite-order torsion obstruction by a scalar phase estimate for divisor masses. This gives uniform Gauduchon mass bounds on each component over compact subsets of the base without assuming torsion-freeness. Bishop compactness then establishes properness of every irreducible component of the relative divisor space, and Barlet's algebraic-dimension theorem yields the conclusion directly. Combined with Rao--Tsai's theorem,the deformation-limit result also shows that every fiber is Moishezon whenever uncountably many fibers are Moishezon. The Fujiki--Pontecorvo families and Huybrechts' twistor and Brauer families clarify the scope and necessity of the hypotheses.

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