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Journal of Mathematics
Volume 2015, Article ID 841746, 7 pages
http://dx.doi.org/10.1155/2015/841746
Research Article

Note on Quasi-Numerically Positive Log Canonical Divisors

Faculty of Education, Gifu Shotoku Gakuen University, Yanaizu-cho, Gifu, Gifu Prefecture 501-6194, Japan

Received 16 January 2015; Accepted 10 April 2015

Academic Editor: Fernando Torres

Copyright © 2015 Shigetaka Fukuda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Kollár, Ed., Flips and Abundance for Algebraic Threefolds, Astérisque 211, 1992.
  2. Th. Bauer, F. Campana, Th. Eckl et al., “A reduction map for nef line bundles,” in Complex Geometry, Goettingen 2000, pp. 27–36, Springer, Berlin, Germany, 2002. View at Google Scholar · View at MathSciNet
  3. F. Ambro, “The moduli b-divisor of an lc-trivial fibration,” Compositio Mathematica, vol. 141, no. 2, pp. 385–403, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, “Existence of minimal models for varieties of log general type,” Journal of the American Mathematical Society, vol. 23, no. 2, pp. 405–468, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. S. Fukuda, “Quasi-numerically positive log canonical divisors,” in Proceedings of the Abstracts of the International Congress of Mathematicians (Short Communications, Poster Sessions), pp. 93–94, Seoul ICM Organizing Committee, Seoul, Republic of Korea, August 2014.
  6. S. Fukuda, “Tsuji's numerically trivial fibrations and abundance,” Far East Journal of Mathematical Sciences, vol. 5, no. 3, pp. 247–257, 2002. View at Google Scholar · View at MathSciNet
  7. H. Tsuji, “Numerical trivial fibrations,” http://arxiv.org/abs/math/0001023.
  8. Y. Kawamata, “Pluricanonical systems on minimal algebraic varieties,” Inventiones Mathematicae, vol. 79, no. 3, pp. 567–588, 1985. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. N. Nakayama, “Invariance of the plurigenera of algebraic varieties under minimal model conjectures,” Topology, vol. 25, no. 2, pp. 237–251, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  10. O. Fujino, “On Kawamata's theorem,” in Classification of Algebraic Varieties, pp. 305–315, European Mathematical Society, Zurich, Switzerland, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  11. M. Reid, “Minimal models of canonical 3-folds,” in Algebraic Varieties and Analytic Varieties, Tokyo 1981, vol. 1 of Advanced Studies in Pure Mathematics, pp. 131–180, North-Holland Publishing, 1983. View at Google Scholar
  12. S. Fukuda, “A base point free theorem of Reid type. II,” Proceedings of the Japan Academy, Series A, Mathematical Sciences, vol. 75, no. 3, pp. 32–34, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  13. S. Fukuda, “An elementary semi-ampleness result for log canonical divisors,” Hokkaido Mathematical Journal, vol. 40, no. 3, pp. 357–360, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. V. Lazic, “Adjoint rings are finitely generated,” http://arxiv.org/abs/0905.2707.
  15. J.-P. Demailly, C. D. Hacon, and M. Păun, “Extension theorems, non-vanishing and the existence of good minimal models,” Acta Mathematica, vol. 210, no. 2, pp. 203–259, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. S. Boucksom, S. Cacciola, and A. F. Lopez, “Augmented base loci and restricted volumes on normal varieties,” Mathematische Zeitschrift, vol. 278, no. 3-4, pp. 979–985, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  17. C. Birkar and Z. Hu, “Log canonical pairs with good augmented base loci,” Compositio Mathematica, vol. 150, no. 4, pp. 579–592, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. S. Cacciola, “On the semiampleness of the positive part of CKM Zariski decompositions,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 156, no. 1, pp. 1–23, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. Y. Kawamata, “On the length of an extremal rational curve,” Inventiones Mathematicae, vol. 105, no. 3, pp. 609–611, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. V.-V. Shokurov, “Letters of a Bi-rationalist. VII Ordered termination,” Proceedings of the Steklov Institute of Mathematics, vol. 264, no. 1, pp. 178–200, 2009. View at Publisher · View at Google Scholar
  21. S. Keel, K. Matsuki, and J. McKernan, “Log abundance theorem for threefolds,” Duke Mathematical Journal, vol. 75, no. 1, pp. 99–119, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, vol. 134 of Cambridge Tracts in Mathematics, Cambridge University Press, 1998.
  23. S. Fukuda, “On log canonical divisors that are log quasi-numerically positive,” Central European Journal of Mathematics, vol. 2, no. 3, pp. 377–381, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  24. F. Ambro, “The locus of log canonical singularities,” http://arxiv.org/abs/math/9806067.
  25. E. Szabó, “Divisorial log terminal singularities,” Journal of Mathematical Sciences, vol. 1, no. 3, pp. 631–639, 1994. View at Google Scholar · View at MathSciNet