Abundance theorem for numerically trivial log canonical divisors of semi-log canonical pairs

Author:
Yoshinori Gongyo

Journal:
J. Algebraic Geom. **22** (2013), 549-564

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.

**[AFKM]**D. Abramovich, L. L. Y. Fong, J. Kollár and J. Kernan, Semi log canonical surface, Flip and Abundance for algebraic threefolds, Astérisque 211 (1992), 139-154.**[A]**Florin Ambro,*The moduli 𝑏-divisor of an lc-trivial fibration*, Compos. Math.**141**(2005), no. 2, 385–403. MR**2134273**, 10.1112/S0010437X04001071**[B1]**Caucher Birkar,*On existence of log minimal models*, Compos. Math.**146**(2010), no. 4, 919–928. MR**2660678**, 10.1112/S0010437X09004564**[B2]**Caucher Birkar,*On existence of log minimal models II*, J. Reine Angew. Math.**658**(2011), 99–113. MR**2831514**, 10.1515/CRELLE.2011.062**[BCHM]**Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan,*Existence of minimal models for varieties of log general type*, J. Amer. Math. Soc.**23**(2010), no. 2, 405–468. MR**2601039**, 10.1090/S0894-0347-09-00649-3**[CKP]**F. Campana, V. Koziarz and M. P un, Numerical character of the effectivity of adjoint line bundles, preprint, arXiv:1004.0584, to appear in Ann. Inst. Fourier.**[CPT]**Frédéric Campana and Thomas Peternell,*Geometric stability of the cotangent bundle and the universal cover of a projective manifold*, Bull. Soc. Math. France**139**(2011), no. 1, 41–74 (English, with English and French summaries). With an appendix by Matei Toma. MR**2815027****[CR]**Charles W. Curtis and Irving Reiner,*Representation theory of finite groups and associative algebras*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Reprint of the 1962 original; A Wiley-Interscience Publication. MR**1013113****[Fj1]**Osamu Fujino,*Abundance theorem for semi log canonical threefolds*, Duke Math. J.**102**(2000), no. 3, 513–532. MR**1756108**, 10.1215/S0012-7094-00-10237-2**[Fj2]**Osamu Fujino,*The indices of log canonical singularities*, Amer. J. Math.**123**(2001), no. 2, 229–253. MR**1828222****[Fj3]**-, Base point free theorems-saturation, b-divisors, and canonical bundle formula-, math.AG/0508554.**[Fj4]**-, On Kawamata's theorem,*Classification of Algebraic Varieties*, 305-315, EMS Ser. of Congr. Rep., Eur. Math. Soc., Zürich, 2010.**[Fj5]**Osamu Fujino,*Fundamental theorems for the log minimal model program*, Publ. Res. Inst. Math. Sci.**47**(2011), no. 3, 727–789. MR**2832805**, 10.2977/PRIMS/50**[Fk1]**Shigetaka Fukuda,*On numerically effective log canonical divisors*, Int. J. Math. Math. Sci.**30**(2002), no. 9, 521–531. MR**1918126**, 10.1155/S0161171202012450**[Fk2]**-, An elementary semi-ampleness result for log canonical divisors, preprint, arXiv:1003.1388.**[G]**Y. Gongyo, On weak Fano varieties with log canonical singularities, preprint, arXiv:0911.0974, to appear in J. Reine Angew. Math.**[Ka1]**Yujiro Kawamata,*Characterization of abelian varieties*, Compositio Math.**43**(1981), no. 2, 253–276. MR**622451****[Ka2]**Y. Kawamata,*Pluricanonical systems on minimal algebraic varieties*, Invent. Math.**79**(1985), no. 3, 567–588. MR**782236**, 10.1007/BF01388524**[Ka3]**Yujiro Kawamata,*Abundance theorem for minimal threefolds*, Invent. Math.**108**(1992), no. 2, 229–246. MR**1161091**, 10.1007/BF02100604**[Ka4]**-, On the abundance theorem in the case of , preprint, arXiv:1002.2682.**[KaMM]**Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki,*Introduction to the minimal model problem*, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR**946243****[KeMM]**Sean Keel, Kenji Matsuki, and James McKernan,*Corrections to: “Log abundance theorem for threefolds” [Duke Math. J. 75 (1994), no. 1, 99–119; MR1284817]*, Duke Math. J.**122**(2004), no. 3, 625–630. MR**2057020**, 10.1215/S0012-7094-04-12236-5

Sean Keel, Kenji Matsuki, and James McKernan,*Log abundance theorem for threefolds*, Duke Math. J.**75**(1994), no. 1, 99–119. MR**1284817**, 10.1215/S0012-7094-94-07504-2**[KoKo]**János Kollár and Sándor J. Kovács,*Log canonical singularities are Du Bois*, J. Amer. Math. Soc.**23**(2010), no. 3, 791–813. MR**2629988**, 10.1090/S0894-0347-10-00663-6**[KoM]**János Kollár and Shigefumi Mori,*Birational geometry of algebraic varieties*, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR**1658959****[NU]**Iku Nakamura and Kenji Ueno,*An addition formula for Kodaira dimensions of analytic fibre bundles whose fibre are Moišezon manifolds*, J. Math. Soc. Japan**25**(1973), 363–371. MR**0322213****[N]**Noboru Nakayama,*Zariski-decomposition and abundance*, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR**2104208****[S]**F. Sakai,*Kodaira dimensions of complements of divisors*, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 239–257. MR**0590433****[Sim]**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****[Siu]**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****[U]**Kenji Ueno,*Classification theory of algebraic varieties and compact complex spaces*, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. MR**0506253**

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

DOI:
https://doi.org/10.1090/S1056-3911-2012-00593-1

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)