Journal of the Mathematical Society of Japan

Log canonical algebras and modules

Caucher BIRKAR

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Let $(X/Z,B)$ be a lc pair with $K_X+B$ pseudo-effective$/Z$ and $Z$ affine. We show that $(X/Z,B)$ has a good log minimal model if and only if its log canonical algebra and modules are finitely generated.

Article information

J. Math. Soc. Japan, Volume 65, Number 4 (2013), 1319-1328.

First available in Project Euclid: 24 October 2013

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Primary: 14E30: Minimal model program (Mori theory, extremal rays)

minimal models log canonical algebra finite generation


BIRKAR, Caucher. Log canonical algebras and modules. J. Math. Soc. Japan 65 (2013), no. 4, 1319--1328. doi:10.2969/jmsj/06541319.

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  • C. Birkar, On existence of log minimal models and weak Zariski decompositions, Math. Ann., 354 (2012), 787–799.
  • C. Birkar, P. Cascini, C. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 23 (2010), 405–468.
  • J.-P. Demailly, C. D. Hacon and M. Păun, Extension theorems, Non-vanishing and the existence of good minimal models, Acta Math., 210 (2013), 203–259.
  • R. Hartshorne, Generalized divisors on Gorenstein schemes, $K$-theory, 8 (1994), 287–339.
  • J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math., 134, Cambridge University Press, 1998.
  • N. Nakayama, Zariski-decomposition and Abundance, MSJ Memoirs, 14, Math. Soc. Japan, Tokyo, 2004.