| Project/Area Number |
15204001
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (A)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | The University of Tokyo |
|---|
Principal Investigator |
KATSURA Toshiyuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor (40108444)
|
|---|
| 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)
森田 康夫 東北大学, 大学院・理学研究科, 教授 (20011653)
森 重文 京都大学, 数理解析研究所, 教授 (00093328)
|
|---|
| Project Period (FY) |
2003 – 2006
|
| Project Status |
Completed (Fiscal Year 2006)
|
| Budget Amount *help |
¥40,040,000 (Direct Cost: ¥30,800,000、Indirect Cost: ¥9,240,000)
Fiscal Year 2006: ¥10,920,000 (Direct Cost: ¥8,400,000、Indirect Cost: ¥2,520,000)
Fiscal Year 2005: ¥9,880,000 (Direct Cost: ¥7,600,000、Indirect Cost: ¥2,280,000)
Fiscal Year 2004: ¥11,830,000 (Direct Cost: ¥9,100,000、Indirect Cost: ¥2,730,000)
Fiscal Year 2003: ¥7,410,000 (Direct Cost: ¥5,700,000、Indirect Cost: ¥1,710,000)
|
|---|
| Keywords | Calabi-Yau manifold / K3 surface / elliptic surface / Abelian varietv / moduli space / Mordell-Weil group / height / Illusie sheaf / 準楕円曲面 / 擬射影平面 / リジッド幾何 / モーデル・ヴェイユ格子 / 代数多様体 / 数論幾何 / 代数幾何 / ピカール数 / アルチン予想 / 符号 / アルチン・メイザー形式群 / 中間ヤコビ多様体 / フェルマー多様体 |
|---|
| 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.
|
