Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T16:33:53.556Z Has data issue: false hasContentIssue false
  • English
  • Français

Log pluricanonical representations and the abundance conjecture

Part of: Birational geometry

Published online by Cambridge University Press:  10 March 2014

Osamu Fujino  and
Yoshinori Gongyo
Osamu Fujino
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan email fujino@math.kyoto-u.ac.jp
Yoshinori Gongyo
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan email gongyo@ms.u-tokyo.ac.jp email y.gongyo@imperial.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

pluricanonical representationabundance conjectureminimal model program

MSC classification

Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 14E07: Birational automorphisms, Cremona group and generalizations
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 4 , April 2014 , pp. 593 - 620
Copyright
© The Author(s) 2014 

References

Abramovich, D. and Karu, K., Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000), 241–273.Google Scholar
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), 139–154.Google Scholar
Ambro, F., Shokurov’s boundary property, J. Differential Geom. 67 (2004), 229–255.CrossRefGoogle Scholar
Birkar, C., On existence of log minimal models II, J. Reine Angew Math. 658 (2011), 99–113.Google Scholar
Birkar, C., Existence of log canonical flips and a special LMMP, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 325–368.Google Scholar
Birkar, C., Log canonical algebras and modules, J. Math. Soc. Japan 65 (2013), 1319–1328.Google Scholar
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), 405–468.Google Scholar
Campana, F., Koziarz, V. and Păun, M., Numerical character of the effectivity of adjoint line bundles, Ann. Inst. Fourier (Grenoble) 62 (2012), 107–119.CrossRefGoogle Scholar
Choi, S. R., The geography of log models and its applications, PhD thesis, Johns Hopkins University (2008).Google Scholar
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.CrossRefGoogle Scholar
Demailly, J.-P., Hacon, C. D. and Păun, M., Extension theorems, non-vanishing and the existence of good minimal models, Acta Math. 210 (2013), 203–259.Google Scholar
Fujino, O., Abundance theorem for semi log canonical threefolds, Duke Math. J. 102 (2000), 513–532.Google Scholar
Fujino, O., Base point free theorem of Reid–Fukuda type, J. Math. Sci. Univ. Tokyo 7 (2000), 1–5.Google Scholar
Fujino, O., The indices of log canonical singularities, Amer. J. Math. 123 (2001), 229–253.Google Scholar
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), 49–62.CrossRefGoogle Scholar
Fujino, O., Introduction to the log minimal model program for log canonical pairs, Preprint (2008), arXiv:0907.1506.Google Scholar
Fujino, O., On Kawamata’s theorem, in Classification of algebraic varieties, EMS Series of Congress Reports (European Mathematical Society, Zürich, 2010), 305–315.Google Scholar
Fujino, O., Semi-stable minimal model program for varieties with trivial canonical divisor, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), 25–30.Google Scholar
Fujino, O., Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), 727–789.Google Scholar
Fujino, O., On isolated log canonical singularities with index one, J. Math. Sci. Univ. Tokyo 18 (2011), 299–323.Google Scholar
Fujino, O., Basepoint-free theorems: saturation, b-divisors, and canonical bundle formula, Algebra Number Theory 6 (2012), 797–823.Google Scholar
Fujino, O., Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), 339–371.Google Scholar
Fujino, O., Fundamental theorems for semi log canonical pairs, Preprint (2012),arXiv:1202.5365.Google Scholar
Fujino, O. and Gongyo, Y., On canonical bundle formulas and subadjunctions, Michigan Math. J. 61 (2012), 255–264.Google Scholar
Fujino, O. and Gongyo, Y., On the moduli b-divisors of lc-trivial fibrations, Preprint (2012),arXiv:1210.5052.Google Scholar
Fujino, O. and Gongyo, Y., On log canonical rings, Adv. Stud. Pure Math., to appear,arXiv:1302.5194.Google Scholar
Fujita, T., Fractionally logarithmic canonical rings of algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), 685–696.Google Scholar
Fukuda, S., On numerically effective log canonical divisors, Int. J. Math. Math. Sci. 30 (2002), 521–531.Google Scholar
Fukuda, S., An elementary semi-ampleness result for log canonical divisors, Hokkaido Math. J. 40 (2011), 357–360.Google Scholar
Gongyo, Y., On the minimal model theory for dlt pairs of numerical log Kodaira dimension zero, Math. Res. Lett. 18 (2011), 991–1000.Google Scholar
Gongyo, Y., On weak Fano varieties with log canonical singularities, J. Reine Angew. Math. 665 (2012), 237–252.Google Scholar
Gongyo, Y., Remarks on the non-vanishing conjecture, Proceeding of Algebraic Geometry in East Asia, Taipei, (2012), to appear.Google Scholar
Gongyo, Y., Abundance theorem for numerically trivial log canonical divisors of semi-log canonical pairs, J. Algebraic Geom. 22 (2013), 549–564.Google Scholar
Hacon, C. D., McKernan, J. and Xu, C., ACC for log canonical thresholds, Preprint (2012),arXiv:1208.4150.Google Scholar
Hacon, C. D. and Xu, C., On finiteness of $B$-representation and semi-log canonical abundance, Preprint (2011), arXiv:1107.4149.Google Scholar
Hacon, C. D. and Xu, C., Existence of log canonical closures, Invent. Math. 192 (2013), 161–195.CrossRefGoogle Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York–Heidelberg, 1977).Google Scholar
Kawamata, Y., Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), 567–588.Google Scholar
Kawamata, Y., Abundance theorem for minimal threefolds, Invent. Math. 108 (1992), 229–246.Google Scholar
Keel, S., Matsuki, K. and McKernan, J., Log abundance theorem for threefolds, Duke Math. J. 75 (1994), 99–119.Google Scholar
Kollár, J., Sources of log canonical centers, in Minimal models and extremal rays, Kyoto, 2011, Adv. Stud. Pure. Math., to appear.Google Scholar
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.Google Scholar
Kollár, J. and Kovács, S. J., Log canonical singularities are Du Bois, J. Amer. Math. Soc. 23 (2010), 791–813.Google Scholar
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.Google Scholar
Lehmann, B., Comparing numerical dimensions Algebra Number Theory 7 (2013), 1065–1100.Google Scholar
Nakayama, N., Zariski-decomposition and abundance, MSJ Memoirs, vol. 4 (Mathematical Society of Japan, Tokyo, 2004).Google Scholar
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), 363–371.Google Scholar
Sakai, F., Kodaira dimensions of complements of divisors, in Complex analysis and algebraic geometry (Iwanami Shoten, Tokyo, 1977), 239–257.CrossRefGoogle Scholar
Ueno, K., Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, vol. 439 (Springer, Berlin, 1975).Google Scholar