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A view on contractions of higher dimensional varieties. (English) Zbl 0948.14014
Kollár, János (ed.) et al., Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9-29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.1), 153-183 (1997).
The article under review provides a compendious exposition of recent developments on studies of the geometric structure of contraction morphisms associated to extremal rays (= Fano-Mori contractions as referred to in the article) of higher dimensional projective algebraic varieties. The notion of extremal rays is introduced by S. Mori [Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)], emerging in an attempt to clarify the mechanism as to how to obtain a minimal model, i.e., an ideal representative of a birational equivalence class whose structure is as natural and irredundant as possible, most befitted to reflect the geometry of the class, in an analogous manner to the classical 2 dimensional case. This program is called the minimal model program, whose idea owes to M. Reid [in: Algebraic Varieties and Analytic Varieties, Proc. Symp., Tokyo 1981, Adv. Stud. Pure Math. 1, 131-180 (1983; Zbl 0558.14028)]. Once the notion of the extremal rays is established, to settle the existence of an associated contraction morphism becomes the next main concern, a special case of which is worked out within the same cited article of Mori, in the 3-dimensional case. There Mori explicitly constructed such a contraction morphism with concrete descriptions of fibers. Most importantly, it is from such description that one realizes the unavoidable occurrence of singularities, a serious evil which hinders the existing methods to search for minimal models in higher dimensions. This makes it necessary to generalize the existence of contraction morphisms to full generality, taking care of the singularities, as one major subsequent focus. This is worked out by Y. Kawamata [Ann. Math., II. Ser. 119, 603-633 (1984; Zbl 0544.14009)], J. Kollár [Ann. Math., II. Ser. 120, 1-5 (1984; Zbl 0544.14010)] and V. V. Shokurov [Math. USSR, Izv. 26, 591-604 (1986); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 3, 635-651 (1985; Zbl 0605.14006)], in arbitrary dimensions, where, unlike Mori’s approach, the proof is in minimal connection with the concrete structure of those contractions but is made rather by such abstract methods as cohomology vanishing [see Y. Kawamata, Math. Ann. 261, 43-46 (1982; Zbl 0476.14007) and E. Viehweg, J. Reine Angew. Math. 335, 1-8 (1982; Zbl 0485.32019)], a generalization of Kodaira’s fundamental theorem.
Meanwhile, there is another focus which, as the course of progress shows, is in every respect as important as the abstract warranty of their mere existence to obtain in-depth knowledge of concrete descriptions of those extremal, or Fano-Mori, contractions. The article under review covers this matter in full detail, which includes, as one highlight, the authors’ own contributions of the generalization of Mori’s structure theorem for birational type contractions from non-singular 3-folds to 4-folds, which also generalizes the idea and method of Y. Kawamata [Math. Ann. 284, No. 4, 595-600 (1989; Zbl 0661.14009)], through the viewpoint of deforming rational curves. To get a good idea about the subject, one has to note the prevalent nature of birational algebraic geometry, that is, to raise the dimension by 1 makes it immensely harder to perform the geometry; there arise unlimited increase of complexities and tremendous ramifications what is not to be distinguished in lower dimensions. This is an utter reality to describe the existing gap between dimension 3 and dimension 4. The authors set forth their systematic study of Fano-Mori contractions in dimension 4, successfully providing a collection of concrete results toward their structures, by introducing several new machineries, most importantly, a methodical application of the dimension formula for the Hilbert scheme of degenerated rational curves. This idea has interacted the investigation of extremal contractions in the reviewer’s article [Y. Kachi, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No. 1, 63-131 (1997; Zbl 0908.14002)], where the same machinery in determining the fiber structures is used for fiber type contractions, inspired by the authors’ idea. The generality of deformations owes to the paper by F. Campana [Ann. Sci. Éc. Norm. Supér., IV. Sér. 25, No. 5, 539-545 (1992; Zbl 0783.14022)] and to Kollár-Miyaoka-Mori’s series of articles [J. Kollár, Y. Miyoka and S. Mori, J. Differ. Geom. 36, No. 3, 765-779 (1992; Zbl 0759.14032), J. Algebr. Geom. 1, No. 3, 429-448 (1992; Zbl 0780.14026) and in: Classification of irregular varieties, minimal models, and abelian varieties, Proc. Conf., Trento 1990, Lect. Notes Math. 1515, 100-105 (1992; Zbl 0776.14012)], where it is introduced the concept of rational chain connection, with which the birational nature of (non-singular) Fano varieties is prescribed, in terms of distribution of algebraic \(1\)-cycles. The striking force of the authors’ success is this concept as their conceptual and strategic guiding principle, based on the fact that Fano varieties appear as (the generic) fibers of Fano-Mori contractions (which by itself is so plain that it tends to be taken for granted). This concept of rational connectivity brings the foremost impact in the study of low-degree algebraic varieties, putting aside its almost equally significant consequence of the boundedness of moduli of Fano varieties, as it affirmatively compensates the failure to rationality of such varieties, in view of the counterexamples to the Lüroth problem, discovered by V. A. Iskovskikh and Yu. I. Manin [Math. USSR, Sb. 15(1971), 141-166 (1972); translation from Mat. Sb., Nov. Ser. 86(128), 140-166 (1971; Zbl 0222.14009)] and C. H. Clemens and P. A. Griffiths [Ann. Math., II. Ser. 95, 281-356 (1972; Zbl 0214.48302)] and others. [As to rational curves of algebraic varieties, the book by J. Kollár, “Rational curves on algebraic varieties” (1995; Zbl 0877.14012) offers a concise and technically complete reference.] The authors thoroughly apply this concept of rational connectivity with technically most elaborate fashion, so as to deduce the analytic structure of arbitrary fibers of Fano-Mori contractions from 4-folds. It should not be too much to stress that to carry the article under review is only possible by multiple accumulation of contemporarily fledged techniques in higher dimensional birational geometry. The article under review shall be acertained to be credited as definitive account for the subject in the whole past decade.
For the entire collection see [Zbl 0882.00032].

14E30 Minimal model program (Mori theory, extremal rays)
14J40 \(n\)-folds (\(n>4\))
14J30 \(3\)-folds
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables