# Limits of Projective and $\partial\bar\partial$-Manifolds under Holomorphic Deformations

Dan Popovici

Arxiv ID: 0910.2032•Last updated: 7/7/2020

We prove that if in a (smooth) holomorphic family of compact complex
manifolds all the fibres, except one, are projective, then the remaining
(limit) fibre must be Moishezon. In an earlier work, we proved this result
under the extra assumption that the limit fibre carries a strongly Gauduchon
metric. In the present paper, we remove the extra assumption by proving that if
all the fibres, except one, are $\partial\bar\partial$-manifolds, then the
limit fibre carries a strongly Gauduchon metric. The
$\partial\bar\partial$-assumption on the generic fibre is much weaker than the
projective, K\"ahler and even {\it class} ${\cal C}$ assumptions, but it
implies the Hodge decomposition and symmetry, while being called the 'validity
of the $\partial\bar\partial$-lemma' by many authors. Our method consists in
starting off with an arbitrary smooth family $(\gamma_t)_{t\in\Delta}$ of
Gauduchon metrics on the fibres $(X_t)_{t\in\Delta}$ and in correcting
$\gamma_0$ in a finite number of steps to a strongly Gauduchon metric by
repeated uses of the $\partial\bar\partial$-assumption on the generic fibre and
of estimates of minimal $L^2$-norm solutions for $\partial$-, $\bar\partial$-
and $d$-equations.

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