Abstract. Let X be a compact Kähler threefold that is not uniruled. We prove that X has a minimal model.

Let X be a compact Kähler threefold with terminal singularities such that K X is nef. We prove that K X is semiample, i.e., some multiple mK X is generated by global sections.

Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent work of Druel and Greb-Guenancia-Kebekus this establishes the Beauville-Bogomolov decomposition for minimal models with trivial canonical class.(a) the reflexive symmetric powers S [l] E are H-stable for every l ∈ N, and (b) the algebraic holonomy group of E (cf. Definition 2.13) is connected.Suppose further that E is pseudoeffective (cf. Definition 2.1). Then c 2 (E) · H n−2 = 0.