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

TL;DR

This paper proves that in a holomorphic family of compact complex manifolds, if all but one fiber are projective or satisfy the ar ext{ extbackslash}partial ext{ extbackslash}bar ext{ extbackslash}d-lemma, then the remaining fiber must be Moishezon, extending previous results without extra assumptions.

Contribution

It removes the extra assumption of a strongly Gauduchon metric on the limit fiber, showing that ar ext{ extbackslash}partial ext{ extbackslash}bar ext{ extbackslash}d-manifolds ensure the limit fiber is Moishezon.

Findings

Limit fiber is Moishezon if all others are projective.

ar ext{ extbackslash}partial ext{ extbackslash}bar ext{ extbackslash}d-manifolds imply the limit fiber has a strongly Gauduchon metric.

Method involves correcting Gauduchon metrics via ar ext{ extbackslash}partial ext{ extbackslash}bar ext{ extbackslash}d assumptions and $L^2$-norm estimates.

Abstract

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.