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Pluricanonical systems on minimal algebraic varieties. (English) Zbl 0593.14010
A minimal algebraic variety X is defined as a normal projective variety having only canonical singularities and whose canonical divisor \(K_ X\) is numerically effective. Such an X is called good if its Kodaira dimension is equal to the numerical Kodaira dimension. The main result in this paper is the following: If X is a good minimal algebraic variety, then the canonical divisor \(K_ X\) is semi-ample. The proof makes use of a vanishing theorem of J. Kollár. The author also discusses a number of conjectures concerning minimal models.
Reviewer: L.D.Olson

14E30 Minimal model program (Mori theory, extremal rays)
14C20 Divisors, linear systems, invertible sheaves
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
Full Text: DOI EuDML
[1] Benveniste X.: Sur l’anneau canonique de certaines variétés de dimension 3. Invent. Math.73, 157-164 (1983) · Zbl 0539.14025
[2] Deligne, P.: Théorie de Hodge III. Publ. Math. IHES.44, 5-78 (1974) · Zbl 0237.14003
[3] Du Bois, P., Jarraud, P.: Une propriété de commutation au changement de base des images directes supérieures du faisceau structural. C. R. Acad. Sc. Paris,279, 745-747 (1974) · Zbl 0302.14004
[4] Elkik, R.: Rationalité des singularités canoniques. Invent. Math.64, 1-6 (1981) · Zbl 0498.14002
[5] Fujita, T.: Zariski decomposition and canonical rings of elliptic threefolds. Preprint (Tokyo-Komaba), 1983 · Zbl 0627.14031
[6] Hartshorne, R.: Residues and Duality. Lecture Notes in Math., vol. 20. Berlin-Heidelberg-New York: Springer 1966 · Zbl 0212.26101
[7] Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. of Math.79, 109-326 (1964) · Zbl 0122.38603
[8] Iitaka, S.: OnD-dimensions of algebraic varieties. J. Math. Soc. Japan23, 356-373 (1971) · Zbl 0212.53802
[9] Kawamata, Y.: Characterization of abelian varieties. Compositio. Math.43, 253-276 (1981) · Zbl 0471.14022
[10] Kawamata, Y.: Kodaira dimension of certain algebraic fiber spaces. J. Fac. Sci. Univ. Tokyo, Sec. IA,30, 1-24 (1983) · Zbl 0516.14026
[11] Kawamata, Y.: A generalization of Kodaira-Ramanujam’s vanishing theorem. Math. Ann.261, 43-46 (1982) · Zbl 0488.14003
[12] Kawamata, Y.: On the finiteness of generators of a pluri-canonical ring for a 3-fold of general type. Amer. J. Math.106, 1503-1512 (1984) · Zbl 0587.14027
[13] Kawamata, Y.: The cone of curves of algebraic varieties. Ann. of Math.119, 603-633 (1984) · Zbl 0544.14009
[14] Kawamata, Y.: Minimal models and the Kodaira dimension of algebraic fiber spaces. Preprint · Zbl 0589.14014
[15] Kollár, J.: Higher direct images of dualizing sheafes. Preprint (Brandeis) 1983
[16] Reid, M.: Canonical 3-folds, in Géométrie, Algébrique Angers 1979, Beauville, A. (ed.), pp. 273-310. Alphen aan den Rijn, The Netherlands: Sijthoff & Noordhoff 1980
[17] Reid, M.: Minimal models of canonical 3-folds. In: Algebraic Varieties and Analytic Varieties, Litaka, S. (ed.) Advanced Studies in Pure Math. vol. 1, pp. 131-180. Kinokuniya, Tokyo, and North-Holland, Amsterdam 1983
[18] Reid, M.: Projective morphisms according to Kawamata. Preprint (Warwick), 1983
[19] Shokurov, V.V.: Theorem on non-vanishing. Preprint (in Russian), 1983
[20] Ueno, K.: Classification Theory of Algebraic Varieties and Compact Complex Space. Lecture Notes in Math., vol. 439. Berlin-Heidelberg-New York: Springer · Zbl 0299.14006
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