Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Artin, Some numerical criteria for contractibility of curves on algebraic surfaces, Amer. J. Math. 84(1962), 485–496.
X. Benveniste, Sur l'anneau canonique de certaines variétés de dimension 3, Invent. Math. 73(1983), 157–164.
E. Bombieri, Canonical models of surfaces of general type, Publ. Math. IHES 42(1973), 171–219.
S. D. Cutkosky, Zariski decomposition of divisors on algebraic varieties, preprint, Brandeis Univ. 1985.
R. Elkik, Rationalité des singularités canoniques, Invent. Math. 64(1981), 1–6.
P. Francia, Some remarks on minimal models I, Compositio Math. 40(1980), 301–313.
A. Fujiki, On the minimal models of complex manifolds, Math. Ann. 253(1980), 111–128.
T. Fujita, Zariski decomposition and canonical rings of elliptic threefolds, preprint, Tokyo Univ. 1984.
_____, A relative version of Kawamata-Viehweg's vanishing theorem, preprint, Tokyo Univ. 1985.
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79(1964), 109–326.
S. Iitaka, On D-dimension of algebraic varieties, J. Soc. Math. Japan 23(1971), 356–373.
Y. Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann. 261(1982), 43–46.
_____, On the finiteness of generators of a pluri-canonical ring for a 3-fold of general type, Amer. J. Math. 106(1984), 1503–1512.
_____, Elementary contractions of algebraic 3-folds, Ann. of Math. 119(1984), 95–110.
_____, The cone of curves of algebraic varieties, Ann. of Math. 119(1984), 603–633.
_____, Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79(1985), 567–588.
_____, Minimal models and the Kodaira dimension of algebraic varieties, preprint, Tokyo Univ. 1985.
_____, The Zariski decomposition of log-canonical divisors, preprint, Tokyo Univ. 1985.
_____, Some properties of minimal algebraic 3-folds, in preparation.
Y. Kawamata, K. Matsuki and K. Matsuda, Introduction to the minimal model problem, in preparation.
S. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. 84(1966), 293–344.
K. Kodaira, Pluricanonical systems on algebraic surfaces of general type, J. Math. Soc. Japan 30(1968), 170–192.
J. Kollár, The cone theorem: Note to [Ka4], Ann. of Math. 120(1984), 1–5.
V. Kulikov, Degenerations of K3 surfaces and Enriques surfaces, Math. USSR Izv. 11(1977), 957–989.
M. Levine, Pluri-canonical divisors on Kähler manifolds, Invent. Math. 74(1983), 293–303.
Y. Miyaoka, Deformations of a morphism along a foliation and applications, preprint, Tokyo Metropolitan Univ. 1985.
_____, The pseudo-effectivity of 3c2 − c1 2 for varieties with numerically effective canonical classes and the non-negativity of the Kodaira dimension of minimal threefolds, preprint, Tokyo Metropolitan Univ. 1985.
Y. Miyaoka and S. Mori, A numerical criterion of uniruledness, preprint.
S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116(1982), 133–176.
A. Moriwaki, Semi-ampleness of the numerically effective part of Zariski decomposition, preprint, Kyoto Univ. 1985.
D. Mumford, The canonical ring of an algebraic surface, (Appendix to [Z]), Ann of Math. 76(1962), 612–615.
N. Nakayama, Invariance of the plurigenera of algebraic varieties under minimal model conjectures, preprint, Tokyo Univ. 1984.
U. Persson and H. Pinkham, Degeneration of surfaces with trivial canonical bundle, Ann. of Math. 113(1981), 45–66.
M. Reid, Canonical 3-folds, in Géométrie Algébriques Angers 1979, A. Beauville ed., 1980, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 273–310.
_____, Minimal models of canonical 3-folds, in Algebraic Varieties and Analytic Varieties, S. Iitaka ed., Advanced Studies in Pure Math. 1(1983), Kinokuniya, Tokyo, and North-Holland, Amsterdam, 131–180.
_____, Projective morphisms according to Kawamata, preprint, Warwick Univ. 1983.
V. V. Shokurov, Theorem on non-vanishing, Math. USSR Izv. 19(1985), ?
_____, On the closed cone of curves of algebraic 3-folds, Math. USSR Izv. 24(1985), 193–198.
_____, Letter to M. Reid, May 24, 1985.
S. Tsunoda, Projective degenerations of algebraic surfaces, in preparation.
K. Ueno, Classification Theory of Algebraic Varieties and Compact Complex Spaces, Lecture Notes in Math. 439(1975), Springer, Berlin-Heidelberg-New York.
E. Viehweg, Vanishing theorems, J. reine angew. Math. 335(1982), 1–8.
P. M. H. Wilson, On the canonical ring of algebraic varieties, Compositio Math. 43(1981), 365–385.
_____, On regular threefolds with x=0, Invent. Math. 76(1984), 345–355.
O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. 76(1962), 560–615.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1986 Springer-Verlag
About this paper
Cite this paper
Kawamata, Y. (1986). On the minimal model problem. In: Grauert, H. (eds) Complex Analysis and Algebraic Geometry. Lecture Notes in Mathematics, vol 1194. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076994
Download citation
DOI: https://doi.org/10.1007/BFb0076994
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16490-6
Online ISBN: 978-3-540-39829-5
eBook Packages: Springer Book Archive