1992 Fiscal Year Final Research Report Summary
Number theory of algebraic varieties
| Project/Area Number |
03452003
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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 | University of Tokyo |
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Principal Investigator |
KAWAMATA Yujiro Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 教授 (90126037)
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| Co-Investigator(Kenkyū-buntansha) |
KUROKAWA Nobushige Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 助教授 (70114866)
SUNADA Toshikazu Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 教授 (20022741)
NAKAMURA Hiroaki Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 助手 (60217883)
NAKAYAMA Noboru Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 助教授 (10189079)
SAITO Takeshi Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 助教授 (70201506)
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| Project Period (FY) |
1991 – 1992
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| Keywords | algebraic varieties / number theory / semistable reduction / minimal model / zeta function / log rithmic structure / Hecke character / Galois representation |
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| Research Abstract |
The purpose of this research was to investigate the number theory of algebraic varieties defined over a ring of algebraic integers. In order to start this investigation, it is important to replace the originalvariety by a more natural model by a birational transformation. If the given variety has relative dimension 1 over the ring of integers, then the classical minimal model theory provides us the canonical model. Kawamata tried to extend the minimal model theory to higher dimensional case, and succeeded in the case in which the relative dimension is 2 and the variety has semistable reduction.
In the course of the proof, newly developed theory of algebraic 3-folds over the complex numbers was used. The difficulty in the proof came from the fact that the vanishing theorem of Kodaira type, which was very useful in the case over the complex numbers, is false in positive characteristic.
The singular fiber of a variety with semistable reduction is a normal crossing variety. Conversely, Kawam
… More
ata considered the smoothing of normal crossing variety into a variety with semistable reduction, and developed the theory of logarithmic deformations with Yoshinori Namikawa at Sophia University. In particular, they proved the existence of a smoothing of a degenerate Calabi-Yau variety.
The cohomology theory is an important tool in the investigaition of algebraic varieties. Saito investigated the 1 dimensional Galois representations on the determinant of L-adic cohomology groups. In the case of constant coefficients, he obtained the description of the corresponding quadratic extensions. In the case of variable coefficients, he proved that they are described by the algebraic Hecke characters determined by the Jacobi sums.
The zeta functions an analytic object which is attached to an algebraic variety over the ring of integers. There are several mysterious conjectures connecting the zeta functions and the number theory of algebraic varieties. Kurokawa investigated multiple zeta funcitons and multiple trigonometric functions, and found formulas of the Gamma factor of the Selberg zeta functions and of the special values of the zeta functions. Less
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