2 - Introduction to the Minimal Model Program

Summary

This chapter provides the first glimpse of the general minimal model program and it also collects some preparatory material.

Section 1 explains the aims and methods of the minimal model program, still at an informal level. One of the fundamental observations is that, starting with dimension three, the minimal model program leads us out of the class of smooth varieties. Therefore, any precise explanation of the minimal model program has to be preceded by a study of the resulting singularities.

Section 2 is an aside; it considers various generalizations of the minimal model program. In applications these are very useful, but they do not introduce new conceptual difficulties.

For us the most useful is the study of the so-called log category. Here one considers pairs (X, D) where X is a variety and D a formal linear combination of irreducible divisors. It seems that for the minimal model program, this provides the natural setting.

Various classes of singularities of such pairs (X, D) are considered in section 3. These are somewhat technical, but indispensable for the later developments. A more detailed study of the log category can be found in [Kol97].

Sections 4 and 5 are devoted to proving the vanishing theorems which are used in subsequent chapters. We prove just as much as needed later, and so we restrict ourselves to the case of smooth projective varieties. In this case the proofs are rather simple and they reveal the relationship of vanishing theorems with the topology of varieties. There are several approaches to vanishing theorems; see [KMM87, EV92, Kol95, Kol97] for other treatments.