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Global generation of adjoint bundles

Published online by Cambridge University Press:  22 January 2016

Hajime Tsuji
Hajime Tsuji*
Affiliation:
Department of Mathematics, Tokyo Institute of Technology 2-12-1 Ohokayama Megro Tokyo, 152, Japan
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In 1988, I. Reider proved that for a smooth projective surface X and an ample line bundle L on X, Kx + 3L is globally generated and Kx + 4L is very ample ([12]). In fact his theorem is much stronger than this (see [12] for detail). Recently a lot of results have been obtained about effective base point freeness (cf. [1, 3, 8, 13, 14, 15]). In particular J. P. Demailly proved that 2KX + 12nnL is very ample for a smooth projective n-fold X and an ample line bundle L on X. [2] will give a good overview for these recent results. The motivation of these works is the following conjecture posed by T. Fujita.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 142 , June 1996 , pp. 5 - 16
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1996

References

[ 1 ] Demailly, J. P., A numerical criterion for very ample line bundles, J. Diff. Geom., 37 (1993), 323–374.Google Scholar
[ 2 ] Demailly, J. P., L vanishing theorem for positive line bundles and adjunction theory, preprint (1994).Google Scholar
[ 3 ] Ein, L. and Lazarsfeld, R., Global generation of pluricanonical and adjoint linear series on smooth projective threefolds, J. Amer. Math. Soc, 6 (1993), 875 — 903.Google Scholar
[ 4 ] Fujita, T., On polarized manifolds whose adjoint bundles are not semipositive, Algebraic Geometry, Sendai, 1985, Advanced Studies Pure Math., Vol. 10, North-Holland, Amsterdam (1987), 167–178.Google Scholar
[ 5 ] Fujita, T., Remarks on Ein-Lazarsfeld criterion of spannedness of adjoint bundles of polarized threefold, preprint (1994).Google Scholar
[ 6 ] Kawamata, Y., The cone of curves of algebraic varieties, Ann. of Math., 119 (1984),603–633.Google Scholar
[ 7 ] Kawamata, Y., Pluricanonical systems of minimal algebraic varieties, Invent. Math., 79, 567–588 (1985).CrossRefGoogle Scholar
[ 8 ] Kobayashi, S. and Ochiai, T., Mappings into complex manifolds with negative first Chern class, J. Math. Soc. Japan, 23 (1971), 137–145.Google Scholar
[ 9 ] Kollar, J., Effective basepoint freeness, Math. Ann., 296 (1993), 595–605.CrossRefGoogle Scholar
[10] Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. of Math., 116 (1982), 133–176.Google Scholar
[11] Nadel, A. M., Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Ann. of Math., 132 (1990), 549–596.Google Scholar
[12] Reider, I., Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math., 127 (1988), 309–316.Google Scholar
[13] Siu, Y. T., An effective Matsusaka’s big theorem, Ann. Inst. Fourier, 43 (1993),1387–1405.Google Scholar
[14] Siu, Y. T., Effective very ampleness, Invent. Math., 124 (1996), 563–571.Google Scholar