Transforms of currents by modifications and 1-convex manifolds

Abstract

One aim of the paper is the study of the strict transform and the total transform of a current T on X via a modification f. Let T be a current of order zero on X: The authors give conditions concerning the existence and the unicity of a strict transform of T. Now, let T be a pluriharmonic (1, 1)-current of order zero on X. For such currents, the authors get sufficient conditions on T to admit a total transform, and prove that this transform is (always) unique. Finally, the main goal of the paper is the following theorem: Let X be a 1-convex manifold of dimension n ≥ 3 and f from X to Y its Remmert reduction, where Y is a Stein quasi-projective space. Let N be a compactification of Y such that N is projective and Sing(N) = Sing(Y ). Let M be a smooth compactification of X. If the map i from H2(X,R) to H2(M,R) induced by the inclusion is injective, then the following properties are equivalent: (a) X is Kähler. (b) X is embeddable. (c) M is projective