Algébricité de l'espace des feuilletages d'un espace analytique compact. (Algebraicity of the space of foliations of a compact analytic space) (Q1107683)

Let X be a reduced compact complex space. Then D. Barlet constructed a moduli space \({\mathcal F}(X)\) of foliations of X (an alternative but different approach is due to G. Pourcin using the Douady space). The author shows that the irreducible components of the closure \(\bar {\mathcal F}(X)\) of \({\mathcal F}(X)\) in the cycle space \(C(G(\theta^ V_ X))\) of the Grassmannian of \(\theta^ V_ X\) are always Moišezon spaces (without assuming X to be Kähler). This result is obtained as a special case of the following Theorem: Let f: \(X\to S\) be a weakly Kähler and surjective morphism between irreducible compact complex spaces with Moišezon fibres. Then the irreducible components of the space \(C_ f(X)\) are Moišezon too. Here \(C_ f(X)\subset C(X)\) is the closure of the Zariski open subset \(C^*_ f(X)\) of C(X), consisting of those cycles which project surjectively to S on each irreducible component. The proof makes use of former results of the author on the structure of weakly Kähler morphisms and of Hironaka's flattening theorem. For the application to foliations, f is even projective where the proof of the main result is somewhat simpler.