2006 Fiscal Year Final Research Report Summary
Arithmetic of Algebraic Varieties and their Moduli Spaces
| Project/Area Number |
15204001
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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 |
Algebra
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| Research Institution | The University of Tokyo |
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Principal Investigator |
KATSURA Toshiyuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor (40108444)
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| Co-Investigator(Kenkyū-buntansha) |
SAITO Takeshi the University of Tokyo, Graduate School of Mathematical Sciences, Professor (70201506)
TERASOMA Tomohide the University of Tokyo, Graduate School of Mathematical Sciences, Professor (50192654)
NAKAMURA Iku Hookaido University, Graduate School of Science, Professor (50022687)
SHIODA Tetsuji Rikkyo University, Department of Science, Professor Emeritus (00011627)
KATO Fumiharu Kyoto University, Graduate School of Science, Associate Professor (50294880)
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| Project Period (FY) |
2003 – 2006
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| Keywords | Calabi-Yau manifold / K3 surface / elliptic surface / Abelian varietv / moduli space / Mordell-Weil group / height / Illusie sheaf |
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| Research Abstract |
Let X be a non-singular complete algebraic variety of dimension n over an algebraically closed field k. X is said to be a Calabi-Yau variety if the canonical bundle is trivial and the cohomology groups of the structure sheaf vanish except for degrees 0 and n. In this research, using the Artin-Mazur formal group of Calabi-Yau variety, I examine the structure of Calabi-Yau variety. As joint-works with van der Geer, I got the following results. Let X^{r} (p) be the Calabi-Yau variety of Fermat type of dimension r in characteristic p>0. Then, we could show the height h of the Artin-Mazur formal group is either one or the infinity, and h is equal to one if and only if p is equal to one modulo r+ 2. M. Artin gave a conjecture that a K3 surface X in characteristic p>0 is supersingular in the sense of Shioda if and only if it is supersingular in the sense of Artin. It is easy to show that for the Fermat K3 surface X^{2} (p) the conjecture holds. Using our results, we see that we cannot generalize the conjecture to the case of higher dimension. We also examined rigid generalized Kummer Calabi-Yau varieties X. The intermediate Jacobian variety of X is isomorphic to an elliptic curve E. We consider the reduction modulo p, and in some special cases we make clear the relation between the height of Artin-Mazur formal group of X mod p and the supersingularity of E mod p. As for the differential forms on Calabi-Yau varieties, we gave a result on the pairing of the cohomology groups of Illusie sheaves. We also showed that the maximal dimension of complete subvariety which is contained in the moduli space M_{2d} of polarized K3 surfaces of degree 2d is equal to 17.
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[Presentation] 正標数ワールド2003
Author(s)
T. Katsura
Organizer
第48回代数学シンポジウム
Place of Presentation
名古屋大学
Year and Date
2003-09-18
Description
「研究成果報告書概要(和文)」より
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[Presentation] 代数幾何学と分類理論2003
Author(s)
T. Katsura
Organizer
日本応用数理学会総合講演
Place of Presentation
京都大学吉田キャンパス
Year and Date
2003-09-18
Description
「研究成果報告書概要(和文)」より
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