Let (X/Z,B+A) be a Q-factorial dlt pair where B,A≥0 are Q-divisors and K X +B+A∼ Q 0/Z. We prove that any LMMP/Z on K X +B with scaling of an ample/Z divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.
@article{PMIHES_2012__115__325_0,
author = {Birkar, Caucher},
title = {Existence of log canonical flips and a special {LMMP}},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {325--368},
publisher = {Springer-Verlag},
volume = {115},
year = {2012},
doi = {10.1007/s10240-012-0039-5},
zbl = {1256.14012},
language = {en},
url = {http://archive.numdam.org/articles/10.1007/s10240-012-0039-5/}
}
TY - JOUR AU - Birkar, Caucher TI - Existence of log canonical flips and a special LMMP JO - Publications Mathématiques de l'IHÉS PY - 2012 SP - 325 EP - 368 VL - 115 PB - Springer-Verlag UR - http://archive.numdam.org/articles/10.1007/s10240-012-0039-5/ DO - 10.1007/s10240-012-0039-5 LA - en ID - PMIHES_2012__115__325_0 ER -
%0 Journal Article %A Birkar, Caucher %T Existence of log canonical flips and a special LMMP %J Publications Mathématiques de l'IHÉS %D 2012 %P 325-368 %V 115 %I Springer-Verlag %U http://archive.numdam.org/articles/10.1007/s10240-012-0039-5/ %R 10.1007/s10240-012-0039-5 %G en %F PMIHES_2012__115__325_0
Birkar, Caucher. Existence of log canonical flips and a special LMMP. Publications Mathématiques de l'IHÉS, Tome 115 (2012), pp. 325-368. doi : 10.1007/s10240-012-0039-5. http://archive.numdam.org/articles/10.1007/s10240-012-0039-5/
[1.] Termination of (many) 4-dimensional log flips, Invent. Math., Volume 168 (2007), pp. 433-448 | DOI | MR | Zbl
[2.] F. Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova, 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhden- nye Algebry, 220–239; translation in Proc. Steklov Inst. Math., 240 (2003), 214–233. | Zbl
[3.] The moduli b-divisor of an lc-trivial fibration, Compos. Math., Volume 141 (2005), pp. 385-403 | DOI | MR | Zbl
[4.] Introduction to Commutative Algebra, Addison-Wesley, Reading, 1969 | Zbl
[5.] On existence of log minimal models, Compos. Math., Volume 146 (2010), pp. 919-928 | DOI | MR | Zbl
[6.] On existence of log minimal models II, J. Reine Angew. Math., Volume 658 (2011), pp. 99-113 | DOI | MR | Zbl
[7.] C. Birkar, Divisorial algebras and modules on schemes, . | arXiv
[8.] Minimal models, flips and finite generation: a tribute to V.V. Shokurov and Y.-T. Siu, Classification of Algebraic Varieties (European Math. Society Series of Congress Reports) (2010) | Zbl
[9.] Existence of minimal models for varieties of log general type, J. Am. Math. Soc., Volume 23 (2010), pp. 405-468 | DOI | MR | Zbl
[10.] Shokurov’s ACC Conjecture for log canonical thresholds on smooth varieties, Duke Math. J., Volume 152 (2010), pp. 93-114 | DOI | MR | Zbl
[11.] Special termination and reduction to pl flips, Flips for 3-Folds and 4-Folds, Oxford University Press, London (2007) | Zbl
[12.] Finite generation of the log canonical ring in dimension four, Kyoto J. Math., Volume 50 (2010), pp. 671-684 | DOI | MR | Zbl
[13.] Semi-stable minimal model program for varieties with trivial canonical divisor, Proc. Jpn. Acad., Ser. A, Math. Sci., Volume 87 (2011), pp. 25-30 | DOI | MR | Zbl
[14.] Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci., Volume 47 (2011), pp. 727-789 | DOI | MR | Zbl
[15.] O. Fujino and Y. Gongyo, Log pluricanonical representations and abundance conjecture, . | arXiv
[16.] O. Fujino and Y. Gongyo, On canonical bundle formulae and subadjunctions, Michigan Math. J. (to appear), . | arXiv | Zbl
[17.] Extension theorems and the existence of flips, Flips for 3-Folds and 4-Folds, Oxford University Press, London (2007) | Zbl
[18.] Existence of minimal models for varieties of log general type II, J. Am. Math. Soc., Volume 23 (2010), pp. 469-490 | DOI | MR | Zbl
[19.] C. D. Hacon and Ch. Xu, Existence of log canonical closures, . | arXiv | Zbl
[20.] C. D. Hacon and Ch. Xu, On finiteness of B-representation and semi-log canonical abundance, . | arXiv
[21.] Algebraic Geometry, Springer, Berlin, 1977 | Zbl
[22.] Log abundance theorem for threefolds, Duke Math. J., Volume 75 (1994), pp. 99-119 | DOI | MR | Zbl
[23.] J. Kollár, Seminormal log centers and deformations of pairs, . | arXiv
[24.] J. Kollár, Which powers of holomorphic functions are integrable?, . | arXiv
[25.] Varieties fibered by good minimal models, Math. Ann., Volume 350 (2011), pp. 533-547 | DOI | MR | Zbl
[26.] Flip theorem and the existence of minimal models for 3-folds, J. Am. Math. Soc., Volume 1 (1988), pp. 117-253 | MR | Zbl
[27.] S. Okawa, On images of Mori dream spaces, . | arXiv
[28.] On the Zariski decomposition problem, Tr. Mat. Inst. Steklova, Volume 240 (2003), pp. 43-73 | MR | Zbl
[29.] Three-dimensional log flips, Russ. Acad. Sci. Izv. Math., Volume 40 (1993), pp. 95-202 (With an appendix in English by Yujiro Kawamata) | DOI | MR | Zbl
[30.] 3-fold log models, Algebr. Geom., 4. J. Math. Sci., Volume 81 (1996), pp. 2667-2699 | MR | Zbl
[31.] Prelimiting flips, Proc. Steklov Inst. Math., Volume 240 (2003), pp. 75-213 | MR | Zbl
[32.] Letters of a bi-rationalist VII: ordered termination, Proc. Steklov Inst. Math., Volume 264 (2009), pp. 178-200 | DOI | MR | Zbl
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