On locally trivial families of analytic subvarieties with locally stable parametrizations of compact complex manifolds (Q2366797)

Let \(Y\) be a compact complex manifold. The author introduces the set \(E(Y)\) of all analytic subvarieties with locally stable parametrizations of \(Y\) and defines \(\tilde Z (Y) \subset Y \times E(Y)\) (respectively \(\overline\pi:\tilde Z(Y) \to E(Y))\) by \(\tilde Z(Y)=\{(y,t)/t \in E(Y),y \in Z_ t\}\) (respectively by restriction to \(\tilde Z(Y)\) of the projection \(Y \times E(Y) \to E(Y))\). The main result announced in this paper establishes that \(E(Y)\) and \(\tilde Z(Y)\) can be endowed with structures of complex separated spaces such that \(\tilde\pi\) becomes analytic, and studies the properties of the triple \((E(Y),\tilde Z(Y),\tilde\pi)\) (in particular the relation between this and the Douady space \(D(Y)\) of \(Y)\).