2003 Fiscal Year Final Research Report Summary
Degeneration of algebraic varieties and log geometry
| Project/Area Number |
14340010
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
KATO Kazuya Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90111450)
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| Co-Investigator(Kenkyū-buntansha) |
MORI Shegefumi Kyoto Univ., Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (00093328)
MARUYAMA Masaki Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50025459)
UENO Kenji Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (40011655)
USHUI Sanpei Osaka Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90117002)
KATO Fumiharu Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50294880)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Hodge structure / Clarifying space / SL(2)-orbit / Log geometry / log abelian variety / ramificaiton theory / L-function / Iwasawa theory |
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| Research Abstract |
1. With S.Usui, I studied classifying spaces of polarized log Hodge structures. We published the paper on the space of SL_2-orbits, and completed the paper "Classifying spaces of degenerating polarized Hodge structures". The last paper has 197 pages in Tex typing, containing various new ideas in log Hodge theory.
Using the new method of log mixed Hodge structures, C.Nakayama, T.Kajiwara and I improved our theory of analytic log abelian varieties and discovered the theory of log complex tori, and wrote joint papers. "Logarithmic abelian varieties, Analytic theory" and "Analytic log Picard varieties".
C.Nakayama, L.Illusie, and I completed a paper "Analytic ket sites" on another study of degeneration of Hodge structures using the analytic Kummer etale site.
2. T.Saito and I found a good ramification theory of schemes by using logarithmic intersection theory, and wrote a paper "Ramification theory for varieties over perfect fields" in which we gave a generalization of Grothendieck-Ogg-Shafarevivch formula to higher dimensions.
3. The log geometry is related to the theory of L-functions via applications to p-adic Hodge theory. In this subject, I wrote a joint paper with T.Fukaya "A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory" and another joint paper with J.Coates, T.Fukaya, R.Sujatha, and O.Venjakob "The GL_2 main conjecture for elliptic curves without complex multiplication" on p-adic zeta functions in non-commutative Iwasawa theory. In these papers, we succeded to formulate the Iwasawa main conjectures in non-commutative Iwasawa theory.
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