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On the Nonemptiness of the Adjoint Linear System of Polarized Manifolds

Published online by Cambridge University Press:  20 November 2018

Yoshiaki Fukuma*
Affiliation:
Department of Mathematics Faculty of Science Tokyo Institute of Technology Oh-okayama, Meguro-ku Tokyo 152 Japan, e-mail: [email protected]
*
Current address: Department of Mathematics College of Education Naruto University of Education Takashima, Naruto-cho, Naruto-shi 772-8502 Japan, e-mail: [email protected]
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Abstract

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Let $(X,L)$ be a polarized manifold over the complex number field with dim$X=n$. In this paper, we consider a conjecture of M. C. Beltrametti and A. J. Sommese and we obtain that this conjecture is true if $n=3$ and ${{h}^{0}}\,(L)\,\ge \,2$, or $\dim\,\text{Bs}|L|\le 0$ for any $n\ge 3$. Moreover we can generalize the result of Sommese.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1998

Footnotes

The author is a Research Fellow of the Japan Society for the Promotion of Science.

References

[BS] Beltrametti, M. C. and Sommese, A. J., The adjunction theory of complex projective varieties. de Gruyter Expositions in Math. 16, Walter de Gruyter, Berlin, New York.Google Scholar
[Fj0] Fujita, T., Classification Theories of Polarized Varieties. London Math. Soc. Lecture Note Series 155 (1990).Google Scholar
[Fj1] Fujita, T., On polarized manifolds whose adjoint bundles are not semipositive. Adv. Stud. Pure Math. 10 (1985), 167178.Google Scholar
[Fj2] Fujita, T., Classification of polarized manifolds of sectional genus two. In: Algebraic Geometry and Commutative Algebra, in Honor of Masayoshi Nagata, Kinokuniya, 1987, 73–98.Google Scholar
[Fk1] Fukuma, Y., A lower bound for the sectional genus of quasi-polarized surfaces. Geom.Dedicata 64 (1997), 229251.Google Scholar
[Fk2] Fukuma, Y., A lower bound for sectional genus of quasi-polarized manifolds. J.Math. Soc. Japan 49 (1997), 339362.Google Scholar
[Fk3] Fukuma, Y., On sectional genus of quasi-polarized 3-folds. Trans. Amer. Math. Soc., to appear.Google Scholar
[I] Ionescu, P., Generalized adjunction and applications. Math. Proc. Cambridge Philos. Soc. 99 (1986), 457472.Google Scholar
[Is] Ishihara, H., On polarized manifolds of sectional genus three. Kodai Math. J. 18 (1995), 328343.Google Scholar
[KMM] Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem. Adv. Stud. Pure Math. 10 (1987), 283360.Google Scholar
[LP] Lanteri, A. and Palleschi, M., About the adjunction process for polarized algebraic surfaces. J. Reine Angew. Math. 352 (1984), 1523.Google Scholar
[So1] Sommese, A. J., Ample divisors on normal Gorenstein surfaces. Abh. Math. Sem. Univ. Hamburg 55 (1985), 151170.Google Scholar
[So2] Sommese, A. J., On the adjunction theoretic structure of projective varieties. In: Proc. Complex Analysis and Algebraic Geometry Conf. 1985, Lecture Notes in Math., Springer, 1986, 175–213.Google Scholar