2004 Fiscal Year Final Research Report Summary
The uniqueness and the degeneracy problems of meromorphic maps and the construction of meromorphic maps with deficient devisors.
| Project/Area Number |
14540196
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Keio University (2004)
Numazu National College of Technology (2002-2003) |
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Principal Investigator |
ATSUJI Atsushi Keio University, Faculty of Economics, Professor, 経済学部, 教授 (00221044)
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| Co-Investigator(Kenkyū-buntansha) |
MORI Seiki Yamagata University, Faculty of Science, Professor, 理学部, 教授 (80004456)
KITAGAWA Yosihisa Utsunomiya University, Faculty of Education, Professor, 教育学部, 教授 (20144917)
KAMADA Hiroyuki Miyagi University of Education, Faculty of Education, Associate Professor, 教育学部, 助教授 (00249799)
AIHARA Yosihiro Numazu National College of Technology, Division of Liberal Arts, Professor, 教養科, 教授 (60175718)
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| Project Period (FY) |
2002 – 2004
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| Keywords | deficiency / meromorphic map / flat tori / Nevanlinna theory / isometric deformation / indefinite Kaehler metric / self duality / delta-subharmonic function |
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| Research Abstract |
Our project was concerned with the value distribution of meromorphic maps,
in particular, uniqueness problem, degeneration and the construction of meromorphic maps with their defect to arbitrary divisors. From 2002-2003 the project leader Aihara and Mori studied the construction of meromorphic maps with positive defect on given hypersurfaces. In particular, we showed that for an arbitrary effective divisor on complex projective spaces there exists an interval of real numbers such that we can always construct the meromorphic maps whose defect equals to each value in the interval. These researches enabled us to improve the estimate on the defect more sharply. We also determined the divisors which defects cannot be evaluated. Aihara also considered some uniqueness problem of meromorphic maps between analytic covering spaces of complex Euclidean spaces and complex projective spaces. He specially discussed which kinds of constraints to the uniqueness are caused by some geometric conditions. From these works we achieved the first aim of our project. We pushed our project forward to some related problems. In 2004, the generalization of the domain of meromorphic maps was considered mainly by the project leader Atsuji. We showed that Nevanlinna theory, the main tool of the theory of value distribution of meromorphic maps, can be established for meromorphic functions on general complete Kaehler manifolds.
We also obtained the following results from 2002-2004 by the effort of the other members.
Kitagawa considered a conjecture : the volume of the domain surrounded by a closed surface in 3-sphere is invariant under any isometric deformations of the closed surface. He obtained the affirmative answer to the conjecture in the case when the closed surface is a flat torus.
Kamada showed that Hirzebruch surfaces which admit Kaehler metrics (may be indefinite) of constant scalar curvature, must be biholomorphic to a direct product of projective lines.
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