Recent developments in the log minimal model program II II Birkar-Cascini-Hacon-McKernan 1 2 2 3 3 5 4 8 4.1.................. 9 4.2.......................... 10 5 11 464-8602, e-mail: fujino@math.nagoya-u.ac.jp 1
1 2 [ 1] 2 [ 2] [ 2] 2 [HM] Shokurov [S2] Siu[Si] [BCHM] [BCHM] [HM] [BCHM] ( Shokurov ) [BCHM] [BCHM] [BCHM] C 1 (Birkar-Cascini-Hacon-McKernan) (X, ) K X + R- π : X U π- K X + π- K X + π- (1) K X + U ( ) 2
(2) K X + π- K X + U (3) K X + Q- m 0 π O X ( m(k X + ) ) O U - [ 2] [ 2] 2 1 [ 2] 3 4 [BOOK] 5 : (A) 17684001 [BCHM] ( ) ( ) 2 [BCHM] 2 X (1) X X X X Q- K X 3
(2) X X X X K X (3) m 0 H0 (X, O X (mk X )) C- (1) (2) (3) 3 (X, ) X X Q- K X + Q- m 0 H0 (X, O X ( m(k X + ) )) [BCHM] [FM] 4 X (X ) m 0 H0 (X, O X (mk X )) 5 (X, ) ϕ : X W (K X + ) ϕ 2 GIT 2 1 f : X S S f 2 0 S f f : X \ f 1 (0) S \ {0} 0 S ( S ) X f 1 (0) X f 1 (0) ( 1)- f 1 (0) ( 1)- f : X S f 1 (0) ( 2)- ( 2)- 4
f : X S K X/S f- X Proj m 0 f O X (mk X/S ) f : X S 0 S f 1 (0) [BCHM] ( 1 (2) ) X S f : X S K X/S f- S Proj m 0 f O X (mk X/S ) S GIT 6 ( ) [BCHM]... [BCHM] [ 2] 3 [ 2] [ 2] 3 ([BCHM] [ 2]) 1 5
1 3 3 3 Viehweg [R] 2 2 n X X S X K X S K S X (K X +S) S = K S X (K X + S) S = K S + Diff S (0) Diff S (0) different Shokurov ([S1] ) S Q- Diff S (0) S X 2 3 3 3 6
II 4 4 X K X X C K X C < 0 Bend and Break Viehweg Bend and Break 2 dim X ( ) ( ) 5 (X, ) 5 X (X, ) Q- Q A + B A Q- B Q- (X, B) NE(X) = NE(X) KX + 0 + R 0 [C] 7
NE(X) KX + 0 NE(X) K X + K X + R 0 [C] K X + Q K X + B + A A K X + K X + K X + (K X + B) Q A K X + 6 6 4.1 4 [BCHM] 60 70 20 [BCHM] [BOOK] 5 5 [HM] [BOOK] 3 4 [BCHM] [BCHM] [BOOK] [K ] [K ] 3 5 6 [K ] 6 7 [BCHM] 8
4.1 [BCHM] [BCHM] (discrepancy) [K ] 2.3 [K ] 3 Viehweg [K ] 4.1 4.2 ( ) [K ] 5 5.1 5.5 5.2 5.4 5.3 [BCHM] [K ] 6 7 6.1 [BCHM] [FA] 16 (adjunction) [BOOK] (1 5 ) [BOOK] 5 Lazarsfeld [L] 9 10 11 (multiplier ideal sheaf) [BOOK] 5 [ 2] [BOOK] [BCHM] [BCHM] [BCHM] 1 2 II [BCHM] 9
[ 2] 4.2 [BOOK] [BOOK] [S2] [BCHM] F. Ambro A. Corti O. Fujino C. Hacon J. Kollár J. McKernan H. Takagi 7 1 Corti 2 Corti 3 pl [S2] 3 pl 3 Fujino 4 Fujino 5 4 2 3 5 Hacon McKernan [HM] [HM] 6 del Pezzo [S2] Shokurov 4 pl Corti McKernan Takagi 7 McKernan Shokurov [HM] 8 Kollár VHS [BCHM] 9 Ambro Shokurov 4 pl 10 Corti 10
5 (special termination) Shokurov 3 ([S1] ) pl Shokurov 3 [FA] 7 [S2] 2 n 1 n Shokurov 4 Prokhorov Prokhorov Prokhorov Shokurov Shokurov Iskovskih ( ) Shokurov Shokurov Iskovskih 11
Shokurov Corti [BOOK] [BOOK] 4 5 [BCHM] [BCHM] [BCHM] Shokurov Kollár Shokurov ( Prokhorov ) [BCHM] C. Birkar, P. Cascini, C. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, arxiv:math/0610203. [BOOK] Flips for 3-folds and 4-folds (Alessio Corti, ed.), Oxford University Press, 2007. [FA] J. Kollár et al., Flips and Abundance for Algebraic Threefolds, Astérisque 211, (1992). 12
[ 1], Recent developments in the log minimal model program ( ), 50, 151 162 (2005). [ 2] [FM],,. O. Fujino, S. Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), no. 1, 167 188. [HM] C. Hacon, J. McKernan, On the existence of flips, math.ag/0507597. [K ] J. Kollár,,,, 1998. [L] [R] [S1] R. Lazarsfeld, Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 49. Springer-Verlag, Berlin, 2004. xviii+385 pp. M. Reid, Young person s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 345 414, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987. V. V. Shokurov, 3-fold log flips, Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95 202 [S2] V. V. Shokurov, Prelimiting flips, Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 82 219; translation in Proc. Steklov Inst. Math. 2003, no. 1 (240), 75 213 [Si] Y.-T. Siu, Invariance of plurigenera, Invent. Math. 134 (1998), no. 3, 661 673. 13