The final chapter treats minimal threefolds. We explain the abundance for threefolds due to Miyaoka and Kawamata depending on the numerical Kodaira dimension. The initial step is to prove the non-vanishing which means the existence of a global section of some pluricanonical divisor. If the irregularity is not zero, then the Albanese map provides enough geometric information. In the case of irregularity zero, Miyaoka applied the generic semi-positivity via positive characteristic. We derive abundance from non-vanishing after replacing the threefold by a special divisorially log terminal pair. Birational minimal models are connected by flops and have the same Betti and Hodge numbers. In dimension three, they have the same analytic singularities. One can expect the finiteness of minimal models ignoring the marking map. This is a part of Kawamata and Morrison's cone conjecture for Calabi-Yau fibrations. We explain Kawamata's work on the conjecture for threefold fibrations with non-trivial base. In dimension three, there exists a uniform number for l such that the l-th pluricanonical map is birational to the Iitaka fibration. We find this number explicitly in the case of general type.

This chapter outlines the general theory of the minimal model program. The program outputs a representative of each birational class which is minimal with respect to the numerical class of the canonical divisor. It grew out of the surface theory with allowing mild singularities. For a given variety, it produces a minimal model or a Mori fibre space after finitely many birational transformations which are divisorial contractions and flips. The program is formulated in the logarithmic framework where we treat a pair consisting of a variety and a divisor. It functions subject to the existence and termination of flips. Hacon and McKernan with Birkar and Cascini proved the existence of flips in an arbitrary dimension. The termination of threefold flips follows from the decrease in the number of divisors with small log discrepancy. Shokurov reduced the termination in an arbitrary dimension to certain conjectural properties of the minimal log discrepancy. It is also important to analyse the representative output by the program. For a minimal model, we expect the abundance which claims the freedom of the linear system of a multiple of the canonical divisor.