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1 Recent developments in the log minimal model program II II Birkar-Cascini-Hacon-McKernan , fujino@math.nagoya-u.ac.jp 1
2 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
![[BCHM] [BCHM] C 1 (Birkar-Cascini-Hacon-McKernan)](/docs-images/42/23044100/images/page_2.jpg "(X, ) K X + R- π : X U π- K X + π- K X + π- (1) K X")
3 (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) [BCHM] ( ) ( ) 2 [BCHM] 2 X (1) X X X X Q- K X 3
![2] 3 4 [BOOK] 5 : (A) 17684001 [BCHM] (](/docs-images/42/23044100/images/page_3.jpg ") ( ) 2 [BCHM] 2 X (1) X X X X Q- K X 3")
4 (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
5 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
6 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
7 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 R 0 [C] 7
8 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 [BCHM] [BCHM] [BOOK] 5 5 [HM] [BOOK] 3 4 [BCHM] [BCHM] [BOOK] [K ] [K ] [K ] 6 7 [BCHM] 8
![1 4 [BCHM] 60 70 20 [BCHM] [BOOK] 5 5 [HM] [BOOK] 3 4](/docs-images/42/23044100/images/page_8.jpg "[BCHM] [BCHM] [BOOK] [K ] [K ] 3 5 6 [K ] 6 7 [BCHM]")
9 4.1 [BCHM] [BCHM] (discrepancy) [K ] 2.3 [K ] 3 Viehweg [K ] ( ) [K ] [BCHM] [K ] [BCHM] [FA] 16 (adjunction) [BOOK] (1 5 ) [BOOK] 5 Lazarsfeld [L] (multiplier ideal sheaf) [BOOK] 5 [ 2] [BOOK] [BCHM] [BCHM] [BCHM] 1 2 II [BCHM] 9
![1 [BCHM] [FA] 16 (adjunction) [BOOK] (1 5 ) [BOOK] 5 Lazarsfeld [L]](/docs-images/42/23044100/images/page_9.jpg "9 10 11 (multiplier ideal sheaf) [BOOK] 5 [ 2] [BOOK] [BCHM] [BCHM]")
10 [ 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 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
![Takagi 7 1 Corti 2 Corti 3 pl [S2] 3 pl 3 Fujino 4 Fujino 5 4 2 3 5 Hacon](/docs-images/42/23044100/images/page_10.jpg "McKernan [HM] [HM] 6 del Pezzo [S2] Shokurov 4 pl Corti McKernan Takagi 7")
11 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
12 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/ [BOOK] Flips for 3-folds and 4-folds (Alessio Corti, ed.), Oxford University Press, [FA] J. Kollár et al., Flips and Abundance for Algebraic Threefolds, Astérisque 211, (1992). 12
13 [ 1], Recent developments in the log minimal model program ( ), 50, (2005). [ 2] [FM],,. O. Fujino, S. Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), no. 1, [HM] C. Hacon, J. McKernan, On the existence of flips, math.ag/ [K ] J. Kollár,,,, [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, xviii+385 pp. M. Reid, Young person s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), , Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, V. V. Shokurov, 3-fold log flips, Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, [S2] V. V. Shokurov, Prelimiting flips, Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, ; translation in Proc. Steklov Inst. Math. 2003, no. 1 (240), [Si] Y.-T. Siu, Invariance of plurigenera, Invent. Math. 134 (1998), no. 3,
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