Gauduchon's form and compactness of the space of divisors

Daniel Barlet (1, 2)

1 IUF - Institut universitaire de France
2 IECL - Institut Élie Cartan de Lorraine

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 ∂ ¯ ∂−lemma, then the algbraic 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.

Mots clés: codimension 1 cycle-space; Family of compact complex manifods; relative; Algebraic dimension; $\partial\bar∂−$lemma

Domaines: Variables complexes [math.CV]; Géométrie algébrique [math.AG]

Daniel Barlet. Gauduchon's form and compactness of the space of divisors. 2017. ⟨hal-01515717v3⟩