# Gauduchon's form and compactness of the space of divisors

## TL;DR

This paper investigates how the algebraic dimension of fibers in a holomorphic family of compact complex manifolds behaves under certain conditions, showing it is lower semi-continuous and preserving properties like being Moishezon.

## Contribution

It establishes that in families where fibers satisfy the -lemma on a dense set, the algebraic dimension cannot decrease, extending understanding of complex manifold deformation.

## Key findings

- Algebraic dimension is lower semi-continuous in the family.
- If fibers are Moishezon on a dense set, all fibers are Moishezon.
- The -lemma condition influences algebraic properties of fibers.

## Abstract

We show that in a holomorphic family of compact complex connected manifolds parametrized by an irreducible complex space $S$, assuming that on a dense Zariski open set $S^{*}$ in $S$ the fibres satisfy the $\partial\bar\partial-$lemma, the algebraic dimension of each fibre in this family is at least equal to the minimal algebraic dimension of the fibres in $S^{*}$. For instance, if each fibre in $S^{*}$ are Moishezon, then all fibres are Moishezon.