1989 Fiscal Year Final Research Report Summary
Problems surrounding manifolds
| Project/Area Number |
63460004
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | Osaka University |
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Principal Investigator |
MIYANISHI Masayoshi Osaka Univ. Fac. Sci., Professor, 理学部, 教授 (80025311)
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| Co-Investigator(Kenkyū-buntansha) |
TSUNODA Shinichiro Osaka Univ. Fac. Sci., Lecturer, 理学部, 講師 (60144424)
SAKANE Yusuke Osaka Univ. Fac. Sci., Ass. Professor, 理学部, 助教授 (00089872)
MURAKAMI Shingo Osaka Univ. Fac. Sci., Professor, 理学部, 教授 (80028068)
KAWAKUBO Katsuo Osaka Univ. Fac. Sci., Professor, 理学部, 教授 (50028198)
KAWANAKA Noriaki Osaka Univ. Fac. Sci., Professor, 理学部, 教授 (10028219)
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| Project Period (FY) |
1988 – 1989
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| Keywords | 3-dimensional cancellation problem / homology plane / algebraic surface of general type / fundamental group / Hecke algebra / manifold of dimension 3 or 4 / Einstein-Kaehler metric / Riemann-Roch theorem |
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| Research Abstract |
During two years of cooperative research, we had steady, noteworthy progress in the following topics.
1. The proposed 3-dimensional cancellation problem was considered in connection with algebraic torus action on the affine space. In the course, we considered homology planes and contractible algebraic surfaces, which the principal investigator succeeded in classifying in the case of Kodaira dimension less than 2. In the case of general type, we found an infinite series of examples of such surfaces.
2. In a study of semi-stable degenerations of algebraic surfaces of general type, Tsunoda and Zhang obtained a logarithmic analogue of the Noether's inequality, with which one is able to develop a theory in the non-complete case corresponding to the one of Horikawa surfaces.
3. The fundamental group of a punctured Riemann surface of positive genus will provide important informations for constructing the Galois extensions of the rational number field. Kaneko made a study on the 1-adic completion of the fundamental group by making use of a filtration by subgroups and clarified the structure of subquotient groups.
4 & 5. Kawanaka worked on the Hecke algebra of a general linear group defined over a finite field. Kawakubo made a new progress in the topological research of Lie group actions on manifolds. Kobayashi obtained new results on manifolds of dimension 3 or 4 through his study on knots.
6. Manifolds with Einstein-Kaehler metrics attract recently lots of attention. Sakane was the first in constructing examples of non-homogeneous Einstein-Kaehler manifolds.
7 & 8. We have made remarkable progress in the theory of global analysis on manifolds. In particular, Ueki gave probabilistic proofs of the big theorems like Riemann-Roch theorem and Gauss-Bonnet theorem.
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