2004 Fiscal Year Final Research Report Summary
HOLOMORPHIC MAPS OF COMPLEX MANIFOLDS
| Project/Area Number |
14540155
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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 | TOKYO INSTITUTE OF TECHNOLOGY |
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Principal Investigator |
TANABE Masaharu TOKYO INSTITUTE OF TECHNOLOGY, Graduate School of Science and Engineering, dept of math, research assistant, 大学院・理工学研究科, 助手 (60272663)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Riemann surfaces / holomorphic mans / theorem of de Franchis |
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| Research Abstract |
Let X be a compact Riemann surface of genus g (> 1). De Franchis stated the following : Theorem of de Franchis. (a) For a fixed compact Riemann surface Y of genus > 1, the number of nonconstant holomorphic maps X → Y is finite. (b) There are only finitely many compact Riemann surfaces Yi of genus > 1 which admit a nonconstant holomorphic map from X. The second statement (b) is often attributed to Seven. After knowing the finiteness of maps, we may ask if there exists a upper bound depending only on some topological invariant, for example, the genus g. Related to the statement (b), the smallest upper bound of the number of maps depending only on the genus compare to the known ones was given.
Also holomorphic maps between compact Riemann surfaces of prime degree was studied. In this case, it was known that if we take suitable homology bases then the matrix representations with respect to the bases are of so called Poinacre normal forms, and the number of such forms is at most the genus. Durin this research it was shown that the number of such forms which actually become representation of maps is just two.
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