Classification of three-dimensional flips

Kollár, János; Mori, Shigefumi

doi:10.1090/S0894-0347-1992-1149195-9

Classification of three-dimensional flips

Authors: János Kollár and Shigefumi Mori
Journal: J. Amer. Math. Soc. 5 (1992), 533-703
MSC: Primary 14E30; Secondary 14B07, 14E05, 14E35, 14J30
DOI: https://doi.org/10.1090/S0894-0347-1992-1149195-9
Erratum: J. Amer. Math. Soc. 20 (2007), 269-271.
MathSciNet review: 1149195
Full-text PDF Free Access

V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. I, Monographs in Mathematics, vol. 82, Birkhäuser Boston, Inc., Boston, MA, 1985. The classification of critical points, caustics and wave fronts; Translated from the Russian by Ian Porteous and Mark Reynolds. MR 777682
Michael Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129–136. MR 199191, DOI https://doi.org/10.2307/2373050
M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277–291. MR 232018, DOI https://doi.org/10.1007/BF01389777
D. C. Spencer and S. Iyanaga (eds.), Global analysis. Papers in honor of K. Kodaira, University of Tokyo Press, Tokyo; Princeton University Press, Princeton, N.J., 1969. MR 0253344

J. Bingener, Lokale Modulräume in den analytischen Geometrie. I-II, Friedr. Vieweg, Braunschweig, 1987.

E. Brieskorn, Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 279–284. MR 0437798
Jan Arthur Christophersen, On the components and discriminant of the versal base space of cyclic quotient singularities, Singularity theory and its applications, Part I (Coventry, 1988/1989) Lecture Notes in Math., vol. 1462, Springer, Berlin, 1991, pp. 81–92. MR 1129026, DOI https://doi.org/10.1007/BFb0086376
Herbert Clemens, János Kollár, and Shigefumi Mori, Higher-dimensional complex geometry, Astérisque 166 (1988), 144 pp. (1989) (English, with French summary). MR 1004926
Akira Fujiki, Deformation of uniruled manifolds, Publ. Res. Inst. Math. Sci. 17 (1981), no. 2, 687–702. MR 642656, DOI https://doi.org/10.2977/prims/1195185269
Hans Grauert, Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math. 15 (1972), 171–198 (German). MR 293127, DOI https://doi.org/10.1007/BF01404124
Hans Grauert, Der Satz von Kuranishi für kompakte komplexe Räume, Invent. Math. 25 (1974), 107–142 (German). MR 346194, DOI https://doi.org/10.1007/BF01390171
Phillip Griffiths and Wilfried Schmid, Recent developments in Hodge theory: a discussion of techniques and results, Discrete subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 31–127. MR 0419850
Alexander Grothendieck, Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.], Secrétariat mathématique, Paris, 1962 (French). MR 0146040

---, Cohomologie local des faisceaux cohérant et théoréme de Lefschetz locaux et globaux - SGA 2, North Holland, Amsterdam, 1968.

Takayuki Hayakawa and Kiyohiko Takeuchi, On canonical singularities of dimension three, Japan. J. Math. (N.S.) 13 (1987), no. 1, 1–46. MR 914313, DOI https://doi.org/10.4099/math1924.13.1
Friedrich Hirzebruch, Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen, Math. Ann. 126 (1953), 1–22 (German). MR 62842, DOI https://doi.org/10.1007/BF01343146
Theo de Jong and Duco van Straten, On the base space of a semi-universal deformation of rational quadruple points, Ann. of Math. (2) 134 (1991), no. 3, 653–678. MR 1135880, DOI https://doi.org/10.2307/2944359
Yujiro Kawamata, On the classification of noncomplete algebraic surfaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., vol. 732, Springer, Berlin, 1979, pp. 215–232. MR 555700
Y. Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), no. 3, 567–588. MR 782236, DOI https://doi.org/10.1007/BF01388524
Yujiro Kawamata, Crepant blowing-up of $3$-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), no. 1, 93–163. MR 924674, DOI https://doi.org/10.2307/1971417

---, Abundance theorem for minimal threefolds, (preprint).

G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518

J. Kollár, Toward moduli of singular varieties, Thesis, Brandeis Univ., 1983.

János Kollár, Toward moduli of singular varieties, Compositio Math. 56 (1985), no. 3, 369–398. MR 814554
János Kollár, Flops, Nagoya Math. J. 113 (1989), 15–36. MR 986434, DOI https://doi.org/10.1017/S0027763000001240

---, Minimal models of algebraic threefolds: Mori’s Program, Asterisque 177-178 (1990), 303-326.

János Kollár, Flips, flops, minimal models, etc, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 113–199. MR 1144527
J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338. MR 922803, DOI https://doi.org/10.1007/BF01389370
Henry B. Laufer, Taut two-dimensional singularities, Math. Ann. 205 (1973), 131–164. MR 333238, DOI https://doi.org/10.1007/BF01350842
Marc N. Levine, Deformations of uni-ruled varieties, Duke Math. J. 48 (1981), no. 2, 467–473. MR 620260
M. Levine, Pluri-canonical divisors on Kähler manifolds, Invent. Math. 74 (1983), no. 2, 293–303. MR 723219, DOI https://doi.org/10.1007/BF01394318
Marc Levine, Pluri-canonical divisors on Kähler manifolds. II, Duke Math. J. 52 (1985), no. 1, 61–65. MR 791292, DOI https://doi.org/10.1215/S0012-7094-85-05205-6
A. Markoe and H. Rossi, Families of strongly pseudoconvex manifolds, Symposium on Several Complex Variables (Park City, Utah, 1970) Springer, Berlin, 1971, pp. 182–207. Lecture Notes in Math., Vol. 184. MR 0304703
T. Matsusaka and D. Mumford, Two fundamental theorems on deformations of polarized varieties, Amer. J. Math. 86 (1964), 668–684. MR 171778, DOI https://doi.org/10.2307/2373030
Yoichi Miyaoka, On the Kodaira dimension of minimal threefolds, Math. Ann. 281 (1988), no. 2, 325–332. MR 949837, DOI https://doi.org/10.1007/BF01458437
Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176. MR 662120, DOI https://doi.org/10.2307/2007050
Shigefumi Mori, On $3$-dimensional terminal singularities, Nagoya Math. J. 98 (1985), 43–66. MR 792770, DOI https://doi.org/10.1017/S0027763000021358
Shigefumi Mori, Flip theorem and the existence of minimal models for $3$-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117–253. MR 924704, DOI https://doi.org/10.1090/S0894-0347-1988-0924704-X
Shigefumi Mori, Birational classification of algebraic $3$-folds, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 307–311. MR 1463707

---, Birational classification of algebraic threefolds, Proc. ICM Kyoto, Springer, Berlin-Heidelberg-New York, 1990.

Noboru Nakayama, Invariance of the plurigenera of algebraic varieties under minimal model conjectures, Topology 25 (1986), no. 2, 237–251. MR 837624, DOI https://doi.org/10.1016/0040-9383%2886%2990042-X

---, The lower semi-continuity of the plurigenera of complex varieties, Algebraic Geometry Sendai (T. Oda, ed.), Adv. Stud. Pure Math., vol. 10, Kinokuniya and North-Holland, 1987, pp. 551-590.

Ziv Ran, Deformations of maps, Algebraic curves and projective geometry (Trento, 1988) Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 246–253. MR 1023402, DOI https://doi.org/10.1007/BFb0085936
Miles Reid, Canonical $3$-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 273–310. MR 605348

---, Minimal models of canonical threefolds, Algebraic Varieties and Analytic Varieties (S. Iitaka, ed.), Adv. Stud. Pure Math., vol. 1, Kinokuniya, Tokyo and North-Holland, Amsterdam, 1983, pp. 131-180.

---, Young person’s guide to canonical singularities, Algebraic Geometry Bowdoin 1985, Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345-416.

V. V. Shokurov, $3$-fold log flips, Izv. Akad. Nauk SSSR Ser. Mat. (to appear).

Yum-tong Siu, The $1$-convex generalization of Grauert’s direct image theorem, Math. Ann. 190 (1970/71), 203–214. MR 279341, DOI https://doi.org/10.1007/BF01433210
J. H. M. Steenbrink, Mixed Hodge structures associated with isolated singularities, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 513–536. MR 713277, DOI https://doi.org/10.1016/0022-4049%2892%2990033-c
J. Stevens, On canonical singularities as total spaces of deformations, Abh. Math. Sem. Univ. Hamburg 58 (1988), 275–283. MR 1027448, DOI https://doi.org/10.1007/BF02941384
Jan Stevens, On the versal deformation of cyclic quotient singularities, Singularity theory and its applications, Part I (Coventry, 1988/1989) Lecture Notes in Math., vol. 1462, Springer, Berlin, 1991, pp. 302–319. MR 1129040, DOI https://doi.org/10.1007/BFb0086390
Jan Stevens, Partial resolutions of rational quadruple points, Internat. J. Math. 2 (1991), no. 2, 205–221. MR 1094706, DOI https://doi.org/10.1142/S0129167X91000144
Bernard Teissier, Variétés toriques et polytopes, Bourbaki Seminar, Vol. 1980/81, Lecture Notes in Math., vol. 901, Springer, Berlin-New York, 1981, pp. 71–84 (French). MR 647489
Jonathan M. Wahl, Equisingular deformations of normal surface singularities. I, Ann. of Math. (2) 104 (1976), no. 2, 325–356. MR 422270, DOI https://doi.org/10.2307/1971049