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Log pluricanonical representations and the abundance conjecture

  • Osamu Fujino (a1) and Yoshinori Gongyo (a2)

Abstract

We prove the finiteness of log pluricanonical representations for projective log canonical pairs with semi-ample log canonical divisor. As a corollary, we obtain that the log canonical divisor of a projective semi log canonical pair is semi-ample if and only if the log canonical divisor of its normalization is semi-ample. We also treat many other applications.

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References

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[AK00]Abramovich, D. and Karu, K., Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000), 241273.
[AFKM92]Abramovich, D., Fong, L.-Y., Kollár, J. and McKernan, J., Semi log canonical surfaces, in Flips and abundance for algebraic threefolds, Astérisque, vol. 211 (Société Mathématique de France, Paris, 1992), 139154.
[Amb04]Ambro, F., Shokurov’s boundary property, J. Differential Geom. 67 (2004), 229255.
[Bir11]Birkar, C., On existence of log minimal models II, J. Reine Angew Math. 658 (2011), 99113.
[Bir12]Birkar, C., Existence of log canonical flips and a special LMMP, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 325368.
[Bir13]Birkar, C., Log canonical algebras and modules, J. Math. Soc. Japan 65 (2013), 13191328.
[BCHM10]Birkar, C., Cascini, P., Hacon, C. and McKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405468.
[CKP12]Campana, F., Koziarz, V. and Păun, M., Numerical character of the effectivity of adjoint line bundles, Ann. Inst. Fourier (Grenoble) 62 (2012), 107119.
[Cho08]Choi, S. R., The geography of log models and its applications, PhD thesis, Johns Hopkins University (2008).
[CR06]Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (AMS Chelsea Publishing, Providence, RI, 2006), reprint of the 1962 original.
[DHP13]Demailly, J.-P., Hacon, C. D. and Păun, M., Extension theorems, non-vanishing and the existence of good minimal models, Acta Math. 210 (2013), 203259.
[Fuj00a]Fujino, O., Abundance theorem for semi log canonical threefolds, Duke Math. J. 102 (2000), 513532.
[Fuj00b]Fujino, O., Base point free theorem of Reid–Fukuda type, J. Math. Sci. Univ. Tokyo 7 (2000), 15.
[Fuj01]Fujino, O., The indices of log canonical singularities, Amer. J. Math. 123 (2001), 229253.
[Fuj07]Fujino, O., What is log terminal?, in Flips for 3-folds and 4-folds, Oxford Lecture Series in Mathematics and its Applications, vol. 35 (Oxford University Press, Oxford, 2007), 4962.
[Fuj08]Fujino, O., Introduction to the log minimal model program for log canonical pairs, Preprint (2008), arXiv:0907.1506.
[Fuj10]Fujino, O., On Kawamata’s theorem, in Classification of algebraic varieties, EMS Series of Congress Reports (European Mathematical Society, Zürich, 2010), 305315.
[Fuj11a]Fujino, O., Semi-stable minimal model program for varieties with trivial canonical divisor, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), 2530.
[Fuj11b]Fujino, O., Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), 727789.
[Fuj11c]Fujino, O., On isolated log canonical singularities with index one, J. Math. Sci. Univ. Tokyo 18 (2011), 299323.
[Fuj12a]Fujino, O., Basepoint-free theorems: saturation, b-divisors, and canonical bundle formula, Algebra Number Theory 6 (2012), 797823.
[Fuj12b]Fujino, O., Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), 339371.
[Fuj12c]Fujino, O., Fundamental theorems for semi log canonical pairs, Preprint (2012),arXiv:1202.5365.
[FG12a]Fujino, O. and Gongyo, Y., On canonical bundle formulas and subadjunctions, Michigan Math. J. 61 (2012), 255264.
[FG12b]Fujino, O. and Gongyo, Y., On the moduli b-divisors of lc-trivial fibrations, Preprint (2012),arXiv:1210.5052.
[FG13]Fujino, O. and Gongyo, Y., On log canonical rings, Adv. Stud. Pure Math., to appear,arXiv:1302.5194.
[Fuj84]Fujita, T., Fractionally logarithmic canonical rings of algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), 685696.
[Fuk02]Fukuda, S., On numerically effective log canonical divisors, Int. J. Math. Math. Sci. 30 (2002), 521531.
[Fuk11]Fukuda, S., An elementary semi-ampleness result for log canonical divisors, Hokkaido Math. J. 40 (2011), 357360.
[Gon11]Gongyo, Y., On the minimal model theory for dlt pairs of numerical log Kodaira dimension zero, Math. Res. Lett. 18 (2011), 9911000.
[Gon12a]Gongyo, Y., On weak Fano varieties with log canonical singularities, J. Reine Angew. Math. 665 (2012), 237252.
[Gon12b]Gongyo, Y., Remarks on the non-vanishing conjecture, Proceeding of Algebraic Geometry in East Asia, Taipei, (2012), to appear.
[Gon13]Gongyo, Y., Abundance theorem for numerically trivial log canonical divisors of semi-log canonical pairs, J. Algebraic Geom. 22 (2013), 549564.
[HMX12]Hacon, C. D., McKernan, J. and Xu, C., ACC for log canonical thresholds, Preprint (2012),arXiv:1208.4150.
[HX11]Hacon, C. D. and Xu, C., On finiteness of $B$-representation and semi-log canonical abundance, Preprint (2011), arXiv:1107.4149.
[HX13]Hacon, C. D. and Xu, C., Existence of log canonical closures, Invent. Math. 192 (2013), 161195.
[Har77]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York–Heidelberg, 1977).
[Kaw85]Kawamata, Y., Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), 567588.
[Kaw92]Kawamata, Y., Abundance theorem for minimal threefolds, Invent. Math. 108 (1992), 229246.
[KMM94]Keel, S., Matsuki, K. and McKernan, J., Log abundance theorem for threefolds, Duke Math. J. 75 (1994), 99119.
[Kol11]Kollár, J., Sources of log canonical centers, in Minimal models and extremal rays, Kyoto, 2011, Adv. Stud. Pure. Math., to appear.
[Kol13]Kollár, J., Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200 (Cambridge University Press, Cambridge, 2013); with the collaboration of S. Kovács.
[KK10]Kollár, J. and Kovács, S. J., Log canonical singularities are Du Bois, J. Amer. Math. Soc. 23 (2010), 791813.
[KM98]Kollár, J. and Mori, S., 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.
[Leh13]Lehmann, B., Comparing numerical dimensions Algebra Number Theory 7 (2013), 10651100.
[Nak04]Nakayama, N., Zariski-decomposition and abundance, MSJ Memoirs, vol. 4 (Mathematical Society of Japan, Tokyo, 2004).
[NU73]Nakamura, I. and Ueno, K., An addition formula for Kodaira dimensions of analytic fibre bundles whose fibre are Moišezon manifolds, J. Math. Soc. Japan 25 (1973), 363371.
[Sak77]Sakai, F., Kodaira dimensions of complements of divisors, in Complex analysis and algebraic geometry (Iwanami Shoten, Tokyo, 1977), 239257.
[Uen75]Ueno, K., Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, vol. 439 (Springer, Berlin, 1975).
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Log pluricanonical representations and the abundance conjecture

  • Osamu Fujino (a1) and Yoshinori Gongyo (a2)

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