This paper is a sequel to the author’s famous paper [Ann. Inst. Fourier 54, No. 3, 499–630 (2004; Zbl 1062.14014)] in which he introduced the theory of geometric orbifolds, that is pairs \((X, \Delta)\) where \(X\) is a normal complex variety (most of the time \(X\) is a compact Kähler or projective manifold) and \(\Delta\) is an effective \(\mathbb Q\)-divisor \(\Delta=\sum (1-\frac{1}{m(D)}) D\) on \(X\) with \(m(D) \geq 1\) being rational or \(\infty\) (most of the time \(m(D)\) is an integer). This definition is basically the same as the definition of a log pair in the minimal model program MMP, but the orbifold divisor \(\Delta\) plays a complete different role. While in the MMP the divisor \(\Delta\) is related to the singularities of (some birational model of) \(X\), the orbifold structure is typically defined in terms of some fibration \(f: Y \rightarrow X\) mapping onto \(X\). Given such a fibration and a prime Weil divisor \(D \subset X\), one defines the multiplicity \(m(D)\) (or more precisely \(m(D, f)\)) as the infimum of the multiplicities \(a_i\) appearing in the decomposition \(f^* D = \sum a_i E_i + R\) where the \(E_i\) map surjectively onto \(D\) and \(R\) does not. In his aforementioned paper the author showed, among many other things, that if one endows the base of a fibration with such an orbifold structure, one obtains a much closer relation between the geometric properties of the total space \(Y\) on the one side and the general fibre \(F\) and the orbifold base \((X, \Delta)\) on the other side. For example we always get an exact sequence of orbifold fundamental groups, up to taking an appropriate bimeromorphic model of the fibration. Another crucial example is given by what the author calls the core fibration: an orbifold is said to be of general type if the canonical divisor \(K_X+\Delta\) is big. A compact Kähler manifold \(Y\) is special if it does not admit any (meromorphic) fibration such that the orbifold base is of general type. The author proved that every compact Kähler manifold \(Y\) admits a unique almost holomorphic fibration \(Y \dashrightarrow X\) such that the base \((X, \Delta)\) is of orbifold general type and the general fibre \(F\) is special. This core fibration is a significant refinement of classical fibrations like the Iitaka fibration which typically has not the property that its base is of general type (in the standard sense).
In the paper under review the author puts his theory on a larger basis by discussing thoroughly the category of geometric orbifolds, the most important points being the appropriate definition of orbifold morphisms and the core fibration in the orbifold setting. He also introduces various notions of rational connectedness by orbifold curves, a very challenging topic with many problems that are open even for surfaces. The final section covers the conjectural relation of special orbifolds with properties from hyperbolic and arithmetic geometry and surveys some of the progress made since the appearance of the first paper. The numerous technical additions and improvements made in this paper should be very helpful for any mathematician interested in this ambitious research program.