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An introduction to the classification of higherdimensional complex varieties. (English) Zbl 0872.14008
Miyaoka, Yoichi et al., Geometry of higher dimensional algebraic varieties. Basel: Birkhäuser. DMV Semin. 26, 129-213 (1997).
These lecture notes provide an introduction to some current aspects of the geometry and classification theory of complex algebraic varieties in dimension at least three. Based on a course that the author taught at a joint seminar with Y. Miyaoka on the topic “Mori theory” [DMV Seminar, Oberwolfach, Germany (April 1995)], the present work and Y. Miyaoka’s lecture notes “Geometry of rational curves on varieties” (in the same volume; see the preceding review) have to be seen as a unit. In fact, both are complementary to each other with respect to the various methodical components of S. Mori’s “minimal model program” for the birational classification theory of complex algebraic varieties. Roughly speaking, the notes under review also emphasize the differential-geometric and complex-analytic viewpoint of the subject, while Y. Miyaoka’s lectures mainly deal with the purely algebro-geometric methods.
More precisely, the author’s aim is to discuss some of those methods and results which allow to describe the position of a projective algebraic manifold within the geography determined by the “algebraic” curvature properties of the canonical bundle. The text consists of eleven sections. After a motivating preface and a brief summary of the necessary prerequisites from basic algebraic geometry, section 0 reviews those parts of the Kodaira classification of smooth projective surfaces which are relevant for the higher-dimensional theory developed in the sequel. Section 1 deals with the different types of singularities occurring in the process of carrying out the minimal model program for a projective manifold with “non-nef” canonical bundle, and section 2 explains some of the cohomological vanishing theorems (due to Kodaira, Kawamata-Viehweg, Nadel, and others) which are usually needed to construct special maps between projective varieties. Elementary properties of ample divisors, the ample cone of a projective variety, and cones of curves (in the sense of S. Mori) are concisely compiled in section 3, while the following section 4 gives detailed accounts on V. Shokurov’s cohomological non-vanishing theorem [Math. USSR, Izv. 26, 591-604 (1986); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 3, 635-651 (1985; Zbl 0605.14006)] and the famous “base point free theorem” for nef divisors on projective varieties with at most terminal singularities. Also, the concepts of canonical models and canonical singularities are derived in this section. The study of projective manifolds with non-nef canonical bundle is continued in section 5, whose main topic is S. Mori’s “cone theorem” [Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)], together with its geometric applications to \(\mathbb{Q}\)-Fano varieties and the existence theorem for rational curves on certain Calabi-Yau manifolds. Section 6 studies Fano manifolds and the structure of contractions of extremal rays (in the sense of S. Mori). – The latter subject is concretely illustrated in section 7, which is devoted to the case of surfaces and threefolds, and which is enhanced by many instructive, explicit examples. In particular, it is shown how a good part of the classical Enriques-Kodaira classification of complex surfaces can be reconstructed by means of these recent methods. Section 8 turns to the “minimal model conjecture”, M. Reid’s “flip conjectures”, their interrelations, and to the (partial) results concerning them. Adjoint bundles to ample divisors on projective varieties and the celebrated Fujita conjecture are discussed in this context, namely in section 9, and the concluding section 10 gives a brief report on threefolds with trivial canonical bundle, with a special emphasis on Calabi-Yau threefolds and the existence of rational curves in them.
Altogether, the author gives a very lucid, mostly very detailed, methodically well-composed and highly instructive state-of-the-art introduction to some of those recent achievements in the birational classification theory of algebraic varieties, which represent the forefront of research in the field and are still in rapid progress. These notes may serve as a valuable guide to the current research literature, in particular to the recent, rather advanced monograph “Flips and abundance for algebraic threefolds” by J. Kollár (ed.) (Summer. Sem., Univ. Utah, Salt Lake City 1991, Astérisque 211 (1992; Zbl 0782.00075)].
For the entire collection see [Zbl 0865.14018].

14E30 Minimal model program (Mori theory, extremal rays)
14F17 Vanishing theorems in algebraic geometry
14J30 \(3\)-folds
14J45 Fano varieties