×

zbMATH — the first resource for mathematics

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

MSC:
14E30 Minimal model program (Mori theory, extremal rays)
14C20 Divisors, linear systems, invertible sheaves
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.