Abundance theorem for numerically trivial log canonical divisors of semi-log canonical pairs
Author:
Yoshinori Gongyo
Journal:
J. Algebraic Geom. 22 (2013), 549-564
DOI:
https://doi.org/10.1090/S1056-3911-2012-00593-1
Published electronically:
November 14, 2012
MathSciNet review:
3048544
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Abstract |
References |
Additional Information
Abstract: We prove the abundance theorem for numerically trivial log canonical divisors of log canonical pairs and semi-log canonical pairs.
References
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- Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208
- F. Sakai, Kodaira dimensions of complements of divisors, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 239–257. MR 0590433
- Carlos Simpson, Subspaces of moduli spaces of rank one local systems, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 3, 361–401. MR 1222278
- Yum-Tong Siu, Abundance conjecture, Geometry and analysis. No. 2, Adv. Lect. Math. (ALM), vol. 18, Int. Press, Somerville, MA, 2011, pp. 271–317. MR 2882447
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References
- D. Abramovich, L. L. Y. Fong, J. Kollár and J. $\mathrm {M^{c}}$Kernan, Semi log canonical surface, Flip and Abundance for algebraic threefolds, Astérisque 211 (1992), 139–154.
- F. Ambro, The moduli $b$-divisor of an lc-trivial fibration, Compositio. Math. 141 (2005), no. 2, 385–403. MR 2134273 (2006d:14015)
- C. Birkar, On existence of log minimal models, Compositio Mathematica 146 (2010), 919–928. MR 2660678
- ---, On existence of log minimal models II, J. Reine Angew Math. 658 (2011), 99–113. MR 2831514
- C. Birkar, P. Cascini, C. D. Hacon and J. $\mathrm {M^{c}}$Kernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405-468. MR 2601039 (2011f:14023)
- F. Campana, V. Koziarz and M. P$\mathrm {\breve {a}}$un, Numerical character of the effectivity of adjoint line bundles, preprint, arXiv:1004.0584, to appear in Ann. Inst. Fourier.
- F. Campana and T. Peternell, Geometric stability of the cotangent bundle and the universal cover of a projective manifold, With an appendix by Matei Toma. Bull. Soc. Math. France 139 (2011), no. 1, 41–74. MR 2815027 (2012e:14031)
- C. W Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Reprint of the 1962 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988. MR 1013113 (90g:16001)
- O. Fujino, Abundance theorem for semi log canonical threefolds, Duke Math. J. 102 (2000), no. 3, 513–532. MR 1756108 (2001c:14032)
- ---, The indices of log canonical singularities, Amer. J. Math. 123 (2001), no. 2, 229–253. MR 1828222 (2002c:14003)
- ---, Base point free theorems–saturation, b-divisors, and canonical bundle formula–, math.AG/0508554.
- ---, On Kawamata’s theorem, Classification of Algebraic Varieties, 305–315, EMS Ser. of Congr. Rep., Eur. Math. Soc., Zürich, 2010.
- ---, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727–789. MR 2832805 (2012h:14031)
- S. Fukuda, On numerically effective log canonical divisors, Int. J. Math. Math. Sci. 30 (2002), no. 9, 521–531. MR 1918126 (2003e:14010)
- ---, An elementary semi-ampleness result for log canonical divisors, preprint, arXiv:1003.1388.
- Y. Gongyo, On weak Fano varieties with log canonical singularities, preprint, arXiv:0911.0974, to appear in J. Reine Angew. Math.
- Y. Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253–276. MR 622451 (83j:14029)
- ---, Pluricanonical systems on minimal algebraic varieties, Inv. Math. 79 (1985), no. 3, 567–588. MR 0782236 (87h:14005)
- ---, Abundance theorem for minimal threefolds. Invent. Math. 108 (1992), no. 2, 229–246. MR 1161091 (93f:14012)
- ---, On the abundance theorem in the case of $\nu =0$, preprint, arXiv:1002.2682.
- Y. Kawamata, K, Matsuda and K, Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, 283–360, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987. MR 946243 (89e:14015)
- S. Keel, K. Matsuki and J. $\mathrm {M^{c}}$Kernan, Log abundance theorem for threefolds. Duke Math. J. 75 (1994), no. 1, 99–119, Corrections to: “Log abundance theorem for threefolds”, Duke Math. J. 122 (2004), no. 3, 625–630. MR 2057020 (2005a:14018); MR 1284817 (95g:14021)
- J. Kollár and S. J Kovács, Log canonical singularities are Du Bois, J. Amer. Math. Soc. 23 (2010), 791–813. MR 2629988
- J. Kollár and S. Mori. Birational geometry of algebraic varieties, Cambridge Tracts in Math., 134 (1998). MR 1658959 (2000b:14018)
- I. Nakamura and K. Ueno, An addition formula for Kodaira dimensions of analytic fibre bundles whose fibre are Moi$\mathrm {\check {s}}$ezon manifolds, J. Math. Soc. Japan 25 (1973), 363–371. MR 0322213 (48:575)
- N. Nakayama, Zarisiki decomposition and abundance, MSJ Memoirs, 14. Mathematical Society of Japan, Tokyo, 2004. MR 2104208 (2005h:14015)
- F. Sakai, Kodaira dimensions of complements of divisors, Complex analysis and algebraic geometry, pp. 239–257. Iwanami Shoten, Tokyo, 1977. MR 0590433 (58:28689)
- C. Simpson, Subspaces of moduli spaces of rank one local systems, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 3, 361–401. MR 1222278 (94f:14008)
- Y. T. Siu, Abundance conjecture, Geometry and analysis. No. 2, 271–317, Adv. Lect. Math. (ALM), 18, Int. Press, Somerville, MA, 2011. MR 2882447
- K. Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Math., Vol. 439, Springer, Berlin, 1975. MR 0506253 (58:22062)
Additional Information
Yoshinori Gongyo
Affiliation:
Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan.
Email:
gongyo@ms.u-tokyo.ac.jp
Received by editor(s):
September 11, 2010
Received by editor(s) in revised form:
March 6, 2011
Published electronically:
November 14, 2012
Additional Notes:
The author was partially supported by the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists (22-7399)