2 A refined Kodaira dimension and its canonical fibration Dedicated to Professor Hirzebruch on his 75 th Birthday

From many earlier works on the classification theory of compact complex manifolds and their intrinsic geometric structures, it has been clear that “positivity” or “nonpositivity” properties of subsheaves of exterior powers of the tangent or cotangent bundle provide some of the most important global information concerning the manifold in question. The programs of Iitaka and Mori on a general classification scheme for algebraic varieties testify to this with a special focus on the top exterior power of the cotangent bundle, the canonical bundle. This paper is motivated by some of our earlier studies to bring the other subsheaves of the cotangent bundle, such as those defined by foliations and fibrations (studied by F. Bogomolov and Y. Miyaoka for example), into focus as important objects of study for a general classification theory in birational geometry. Especially relevant here are the line subsheaves of exterior powers of the cotangent bundle that correspond to the canonical “bundles” of the orbifold bases of fibrations.