anos Kollar, Shigefumi Mori Birational Geometry of Algebraic Varieties
The minimal model program, or Mori's program, was one of the great successes of algebraic geometry in the 1980s. The basic goal was to understand the birational geometry of threefolds in a way analogous to the birational theory of surfaces. This approach got its start with work of Mori, who began to study the role of curves on threefolds X which intersected the canonical class of KX negatively. This led to a generalisation of the notion of minimal model in dimension 2. A (non-singular) projective variety X is minimal if KX is nef, i.e. KX⋅C≥0 for any effective curve C on X. An algorithm for finding minimal models emerged. Mori showed that if KX was not nef, X necessarily contained a rational curve which could be contracted. These rational curves would not be isolated, but rather all curves numerically equivalent to the one being contracted would also be contracted. This yields a morphism f:X→Y, which might either give X a fibre space structure (now known as a Mori fibre space) for which −KX is relatively ample, or else f would be a birational contraction. Unfortunately, Y may then be singular, and this program had to be extended to include varieties with certain mild classes of singularities (terminal, canonical, log-terminal,…). The basic results which enable one to carry out the minimal model program and show the existence of such contractions were proven in this more general context through the efforts of a large number of mathematicians. One difficulty remained. Sometimes the contractions f:X→Y might be birational but with exceptional locus E having codimension at least 2. In this case there is a catastrophic breakdown of the program, in the sense that Y does not have singularities in which one is able to make sense of the program. A flip is necessary. This is a birational map g:X⇢X+ defined over Y. If the exceptional locus of X+→Y is E+, then g:X−E→X+−E+ is an isomorphism, and codim(E+)≥2. Furthermore, π+:X+→Y should not have the property that KX+ is relatively ample. The proof of the existence of flips in dimension three was the hardest part of the program and was completed by Mori in 1988. Existence in higher dimensions is still unknown. The book under review, written by two of the leaders in the field, is a comprehensive treatment of the minimal model program. The text strives to be self-contained and to give complete proofs; the level of knowledge needed is that of [R. Hartshorne, Algebraic geometry, Springer, New York, 1977; MR0463157]. The book begins with a survey of the earlier techniques (the method of bend and break) used in the theory, which enables one to prove many of the important theorems in the non-singular case. In the second chapter, the many different flavours of singularities which have been introduced by minimal model theorists are defined. In the third chapter, the cone theorems are proved in complete generality. At this point, modulo flips, one is ready to run the minimal model program. The fourth and fifth chapters go into more detail on the types of singularities under discussion, with surface singularities and results about simultaneous resolutions treated in the fourth chapter and threefold singularities in the fifth chapter. The sixth chapter deals with threefold flops. These are birational operations analogous to flips, and are associated to small contractions f:X→Y for which KX is relatively trivial. These operations get at the issue of non-uniqueness of minimal models of threefolds: two birational minimal models must differ by a sequence of flops. Existence of flops for threefolds with canonical singularities is proven, as well as the existence of Q-factorizations. The last chapter deals with semistable flips, and proves their existence. This is a subclass of all flips, whose existence was proven before the general case, the latter being far too technical and lengthy for a textbook. Finally the chapter ends with a brief survey on other work involving threefolds and higher-dimensional birational geometry. The beginning reader might find the going rather technical at times, and might want to start with the earlier text by H. Clemens, Kollár and Mori [Astérisque No. 166 (1988), 144 pp. (1989); MR1004926], from which the first chapter of this text is drawn. Nevertheless, the text under review will prove invaluable for the more advanced student of the minimal model program, as well as researchers in the field.
Reviewer: Gross, Mark [form MathSciNet]