Abstract : Let f : X → Y be a holomorphic map between compact Kähler manifolds. If a general fibre of f is a projective manifold a natural question is whether the morphism itself is projective, i.e. X embeds into some projectivised bundle P(V) → Y. It is well-known that this is not the case, but we will see that in some situations that are natural in the context of MMP, the answer is yes.

Keywords : Kähler morphisms; projectivity criteria; MMP; projective morphism