2000 Fiscal Year Final Research Report Summary
Radon transformation in Nevanlinina theory and Diophantire approximation
| Project/Area Number |
09304007
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| Research Category |
Grant-in-Aid for Scientific Research (A).
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Nagoya University |
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Principal Investigator |
KOBAYASHI Ryoichi Graduate School of Mathematics, Nagoya University, Professor, 大学院・多元数理科学研究科, 教授 (20162034)
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| Co-Investigator(Kenkyū-buntansha) |
NOGUCHI Junjiro Graduate School of Mathematical Science, Nagoya University, Professor, 大学院・数理科学研究科, 教授 (20033920)
SATO Takeshi Graduate School of Mathematics, Nagoya University, Assistant Professor, 大学院・多元数理科学研究科, 助手 (60252219)
OHSAWA Takeo Graduate School of Mathematics, Nagoya University, Professor, 大学院・多元数理科学研究科, 教授 (30115802)
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| Project Period (FY) |
1997 – 2000
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| Keywords | Nevanlinna theory / Diophantine geometry / Lemma on logauthmic derivative / integral geometry / Radon transformation |
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| Research Abstract |
The second main conjective is the ultimate purpose of the Nevanlinna theory. In this research, I proposed several ideas of the "conjectural geometry" unifying Nevanlinna theory and Diophantive approximation through the attempt toward the second main conjecture. The analogy between Castan-Ahlfors-Weyl theory of holomorphic curves in Pn and the sulpace theorem of Schmidt on Diophantino approximation on projective spaces is ohsewed not only in their statements but also in their profs. The essence of the analogy lies in the idea of the "Radon-transformation" in integral geomeyty, In this research, I was able to establish a kind of "Radon-transform" which lies behind both Nevanlinna theory and Diophantine approximation. One advantage of this "Radon-transform" is the following : through this transformation we will be able to separate the part of the Diophantine geometry which is completely analogous to the part of Nevanlinna theory in which the lemma on logarithmic derivative plays an essent
… More
ial role. This part in Diophantine geometry is most number-theoretic and the theorems of Diophantine approximation of Roth-Schmidt-Faltings type, as well as Mihkowski-Bombieri-Vasler "geometry of numbers" play the most enential role. This is most different from holomorphic case, but never the less the result is completely analogous to the lemnia on loganithmic derivature. On the other hand, there was a big progress on the attempt toward 2nd main conjecture. Namely I discovered the geometric framework of n independent commutative holomorphic vector fields and the singularity caused by introducing such vector fields on "noncommutative" projective varieties. The wronskian formalism is possible in terms of these n commutative vector fields. As a result, the difficulty is localized in "singularities" and I was able to establish a new lemma on loganthmic derivative ("projective version") to analyze holomorphic curves in the presence of "singularities". I hope this will provide a new method in complex algebraic geometry. Less
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