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Polarized pairs, log minimal models, and Zariski decompositions

Published online by Cambridge University Press:  11 January 2016

Caucher Birkar  and
Zhengyu Hu
Caucher Birkar
Affiliation:
DPMMS, Centre for Mathematical Sciences, Cambridge University, Cambridge CB3 0WB, United Kingdom, c.birkar@dpmms.cam.ac.uk
Zhengyu Hu
Affiliation:
DPMMS, Centre for Mathematical Sciences, Cambridge University, Cambridge CB3 0WB, United Kingdom, zh262@dpmms.cam.ac.uk
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Abstract

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We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let (X,B) be a projective log canonical pair. We will show that (X,B) has a log minimal model if either KX + B birationally has a Nakayama–Zariski decomposition with nef positive part, or if KX +B is big and birationally has a Fujita–Zariski or Cutkosky–Kawamata–Moriwaki–Zariski decomposition. Along the way we introduce polarized pairs (X,B +P), where (X,B) is a usual projective pair and where P is nef, and we study the birational geometry of such pairs.

Keywords

14E30
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 215 , September 2014 , pp. 203 - 224
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[1] Birkar, C., On existence of log minimal models, II, J. Reine Angew. Math. 658 (2011) 99–113. MR 2831514. DOI 10.1515/CRELLE.2011.062.Google Scholar
[2] Birkar, C., Existence of log canonical flips and a special LMMP, Publ. Math. Inst. Hautes Etudes Sci. 115 (2012), 325–368. MR 2929730. DOI 10.1007/s10240–012-0039–5.Google Scholar
[3] Birkar, C., On existence of log minimal models and weak Zariski decompositions, Math. Ann. 354 (2012), 787–799. MR 2965261. DOI 10.1007/s00208–011-0756-y.Google Scholar
[4] Birkar, C. and Chen, J. A., Varieties fibred over abelian varieties with fibres of log general type, preprint, arXiv:1311.7396v1 [math.AG].Google Scholar
[5] Birkar, C. and Chen, Y., Images of manifolds with semi-ample anti-canonical divisor, to appear in J. Algebraic Geom., preprint, arXiv:1207.4070v1 [math.AG].Google Scholar
[6] Cacciola, S., On the semiampleness of the positive part of CKM Zariski decompositions, Math. Proc. Cambridge Philos. Soc. 156 (2014), 1–23. MR 3144208. DOI 10.1017/S0305004113000509.Google Scholar
[7] Campana, F., Chen, J. A., and Peternell, T., Strictly nef divisors, Math. Ann. 342 (2008), 565–585. MR 2430991. DOI 10.1007/s00208–008-0248-x.Google Scholar
[8] Fujino, O. and Gongyo, Y., Log pluricanonical representations and abundance conjecture, preprint, arXiv:1104.0361v3 [math.AG].Google Scholar
[9] Fujino, O. and Mori, S., A canonical bundle formula, J. Differential Geom. 56 (2000), 167–188. MR 1863025.Google Scholar
[10] Fujita, T., Zariski decomposition and canonical rings of elliptic threefolds, J. Math. Soc. Japan 38 (1986), 19–37. MR 0816221. DOI 10.2969/jmsj/03810019.Google Scholar
[11] Kawamata, Y., “The Zariski decomposition of log-canonical divisors” in Algebraic Geometry, Bowdoin, 1985, Pt. 1 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., Providence, 1987, 425–433. MR 0927965.Google Scholar
[12] Kawamata, Y., On the length of an extremal rational curve, Invent. Math. 105 (1991), 609–611. MR 1117153. DOI 10.1007/BF01232281.Google Scholar
[13] Kawamata, Y., Subadjunction of log canonical divisors, II, Amer. J. Math. 120 (1998), 893–899. MR 1646046.Google Scholar
[14] Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge University Press, Cambridge, 1998. MR 1658959. DOI 10.1017/CBO9780511662560.Google Scholar
[15] Lehmann, B., On Eckl’s pseudo-effective reduction map, Trans. Amer. Math. Soc. 366, no.3 (2014), 1525–1549. MR 3145741.Google Scholar
[16] Moriwaki, A., Semiampleness of the numerically effective part of the Zariski decomposition, J. Math. Kyoto Univ. 26 (1986), 465–481. MR 0857230.Google Scholar
[17] Nakayama, N., Zariski-Decomposition and Abundance, MSJ Mem. 14, Math. Soc. Japan, Tokyo, 2004. MR 2104208.Google Scholar
[18] Prokhorov, Y. G., On the Zariski decomposition problem (in Russian), Tr. Mat. Inst. Steklova 240 (2003), 43–72; English translation in Proc. Steklov Inst. Math. 240 (2003), 37–65. MR 1993748.Google Scholar
[19] Shokurov, V. V., 3-fold log models, J. Math. Sci. (N.Y.) 81 (1996), 2667–2699. MR 1420223. DOI 10.1007/BF02362335.Google Scholar
[20] Shokurov, V. V., Complements on surfaces, J. Math. Sci. (N.Y.) 102 (2000), 3876–3932. MR 1794169. DOI 10.1007/BF02984106.Google Scholar
[21] Shokurov, V. V., Letters of a bi-rationalist, VII: Ordered termination (in Russian), Tr. Mat.Inst. Steklova 264 (2009), 184–208; English translation in Proc. Steklov Inst. Math. 264 (2009), 178–200. MR 2590847. DOI 10.1134/S0081543809010192.Google Scholar