The first main theorem on complements: from global to local

Yu. G. Prokhorov ; V. V. Shokurov

Izvestiya. Mathematics , Tome 65 (2001) no. 6, pp. 1169-1196.

Résumé

The purpose of this paper is to explain and generalize the methods of [24] (see also [18] and [19]). We establish that for local Fano contractions the existence of complements can be reduced to the existence of complements for projective Fano varieties of smaller dimension.

Bibliographie

[1] Abe T., Classification of Exceptional Complements: Elliptic Curve Case, E-print alg-geom/9711029

[2] Alexeev V., “Boundedness and $K^2$ for log surfaces”, Int. J. Math., 5 (1994), 779–810 | DOI | MR | Zbl

[3] Borisov A. A., “Boundedness theorem for Fano log-threefolds”, J. Algebraic Geom., 5 (1996), 119–133 | MR | Zbl

[4] Borisov A. A., Borisov L. A., “Osobye toricheskie mnogoobraziya Fano”, Matem. sb., 183:2 (1992), 134–141 | MR | Zbl

[5] Fujino O., “Abandance theorem for semi log canonical threefolds”, Duke Math. J., 102:3 (2000), 513–532 | DOI | MR | Zbl

[6] Ishii S., “The quotients of log-canonical singularities by finite groups”, Singularities, Sapporo, 1998; Adv. Stud. Pure Math., 29 (2000), 135–161 | Zbl

[7] Ishii S., Prokhorov Yu. G., “Hypersurface exceptional singularities”, International J. Math., 12:5 (2001), 1–27 ; E-print math.AG/9910123 | MR

[8] Kawamata Y., Matsuda K., Matsuki K., “Introduction to the minimal model program”, Adv. Stud. Pure Math., 10 (1987), 283–360 | MR | Zbl

[9] Keel S., McKernan J., “Rational curves on quasi-projective surfaces”, Memoirs AMS, 140:669 (1999) | MR

[10] Kollár J., “Log surfaces of general type; some conjectures”, Contemporary Math. AMS, 162 (1994), 261–275 | MR | Zbl

[11] Kollár J., “Singularities of pairs”, Proc. Symp. Pure Math., 62 (1995), 221–287 | MR

[12] Kollár J., Mori S., “Classification of three dimensional flips”, J. Amer. Math. Soc., 5 (1995), 533–703 | DOI | MR

[13] Kollár J. et al., “Flips and abundance for algebraic threefolds”, A summer seminar at the University of Utah (Salt Lake City. 1991), Astérisque, 211, 1992 | MR

[14] Markushevich D., Prokhorov Yu. G., “Exceptional quotient singularities”, Amer. J. Math., 121 (1999), 1179–1189 | DOI | MR | Zbl

[15] Mori S., “Flip theorem and the existence of minimal models for $3$-folds”, J. Amer. Math. Soc., 1 (1988), 117–253 | DOI | MR | Zbl

[16] Nikulin V. V., “Poverkhnosti del Petstso s logterminalnymi osobennostyami, I”, Matem. sb., 180 (1989), 226–243 | MR | Zbl

[17] Nikulin V. V., “Poverkhnosti del Petstso s logterminalnymi osobennostyami, III”, Izv. AN SSSR. Ser. matem., 53:6 (1989), 1316–1334 | MR

[18] Prokhorov Yu. G., “Blow-ups of canonical singularities”, Algebra, Proc. Internat. Algebraic Conf. (Moscow, Russia, May 25–30, 1998), ed. Yu. Bahturin, Walter der Greither Publ., Berlin, 2000, 301–317 | MR | Zbl

[19] Prokhorov Yu. G., “Boundedness of nonbirational extremal contractions”, Internat. J. Math., 11 (2000), 393–411 | MR | Zbl

[20] Prokhorov Yu. G., Lectures on complements on log surfaces, MSJ Memoirs, 10, Math. Soc. of Japan, 2001 | MR

[21] Prokhorov Yu. G., “Ogranichennost isklyuchitelnykh faktorosobennostei”, Matem. zametki, 68:5 (2000), 786–789 | MR | Zbl

[22] Shepherd-Barron N. I., Wilson P. M. H., “Singular threefolds with numerically trivial first and second Chern classes”, J. Algebraic Geom., {3} (1994), 265–281 | MR | Zbl

[23] Shokurov V. V., “Trekhmernye logperestroiki”, Izv. AN SSSR. Ser. matem., 56:1 (1992), 105–203 | MR | Zbl

[24] Shokurov V. V., “Complements on surfaces”, J. Math. Sci., 102:2 (2000), 3876–3932 | DOI | MR | Zbl

[25] Shokurov V. V., “$3$-fold log models”, J. Math. Sci., 81 (1996), 2667–2699 | DOI | MR | Zbl

[26] Shokurov V. V., Pl flips, Preprint, Baltimore–Moscow, 2000 | MR | Zbl