Flips, flops, minimal models, etc.

*(English)*Zbl 0755.14003
Proc. Conf., Cambridge/MA (USA) 1990, Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 113-199 (1991).

[For the entire collection see Zbl 0743.00018.]

There exist already some survey articles on the minimal model program [e.g. Y. Kawamata, K. Matsuda and K. Matsuki in Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 283–360 (1987; Zbl 0672.14006) or the author, Bull. Am. Math. Soc., New Ser. 17, 211–273 (1987; Zbl 0649.14022)], but this paper does not reproduce them. Only in the first chapters the author presents the main results in Mori’s program mostly without proofs. Then he studies in more detail flips and flops. There are “surgery type” operations changing a threefold only in codimension two and showing the difference to the classification theory of surfaces. After discussing several applications of the minimal model program in dimension three he tries to answer the question how to find extremal rays. For that purpose he introduces “seemingly extremal rays” which seem to be the correct generalization of the notion of extremal rays for nonprojective threefolds.

At several places in the whole paper and especially in \(\S5\) he investigates nonprojective (but compact Moishezon) threefolds with the techniques of Mori’s program. For example he proves that a Moishezon threefold being homeomorphic to \(\mathbb P^ 3\) is actually isomorphic to it. In the final chapter he shows that classification theory in dimension three is closely related to the theory of deformations of rational surface singularities. The article is nicely to be read thanks to many examples illustrating the notions introduced and the difficulties appearing. Various conjectures and open problems related to minimal model theory are presented.

There exist already some survey articles on the minimal model program [e.g. Y. Kawamata, K. Matsuda and K. Matsuki in Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 283–360 (1987; Zbl 0672.14006) or the author, Bull. Am. Math. Soc., New Ser. 17, 211–273 (1987; Zbl 0649.14022)], but this paper does not reproduce them. Only in the first chapters the author presents the main results in Mori’s program mostly without proofs. Then he studies in more detail flips and flops. There are “surgery type” operations changing a threefold only in codimension two and showing the difference to the classification theory of surfaces. After discussing several applications of the minimal model program in dimension three he tries to answer the question how to find extremal rays. For that purpose he introduces “seemingly extremal rays” which seem to be the correct generalization of the notion of extremal rays for nonprojective threefolds.

At several places in the whole paper and especially in \(\S5\) he investigates nonprojective (but compact Moishezon) threefolds with the techniques of Mori’s program. For example he proves that a Moishezon threefold being homeomorphic to \(\mathbb P^ 3\) is actually isomorphic to it. In the final chapter he shows that classification theory in dimension three is closely related to the theory of deformations of rational surface singularities. The article is nicely to be read thanks to many examples illustrating the notions introduced and the difficulties appearing. Various conjectures and open problems related to minimal model theory are presented.

Reviewer: B.Kreußler (Kaiserslautern)

##### MSC:

14E30 | Minimal model program (Mori theory, extremal rays) |

14J30 | \(3\)-folds |

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

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\textit{J. Kollár}, in: Surveys in differential geometry. Vol. I: Proceedings of the conference on geometry and topology, held at Harvard University, Cambridge, MA, USA, April 27-29, 1990. Providence, RI: American Mathematical Society; Bethlehem, PA: Lehigh University. 113--199 (1991; Zbl 0755.14003)