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Arithmetic of Algebraic Varieties and their Moduli Spaces

代数簇及其模空间的算术

基本信息

批准号：
15204001
负责人：
KATSURA Toshiyuki
金额：
$ 25.63万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

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.

设X是代数闭域k上的n维非奇异完备代数簇。称X是卡-丘簇，如果标准丛是平凡的，且结构层的上同调群除了0度和n度外为零。本文利用Calabi-Yau簇的Artin-Mazur形式群，研究了Calabi-Yau簇的结构。作为与货车der Geer的联合工作，我得到了以下结果。设X^{r}（p）是特征p>0的维数为r的Fermat型Calabi-Yau簇。然后，我们可以证明Artin-Mazur形式群的高度h是1或无穷大，并且h等于1当且仅当p等于1模r+ 2。M. Artin猜想特征p>0的K3曲面X是Shioda意义下的超奇异曲面当且仅当它是Artin意义下的超奇异曲面.很容易证明，对于费马K3曲面X^{2}（p），该猜想成立。利用我们的结果，我们看到，我们不能推广的情况下，更高的维猜想。我们还研究了刚性广义库默Calabi-Yau簇X. X的中间雅可比簇同构于椭圆曲线E。考虑了模p的约化，在某些特殊情况下，明确了X mod p的Artin-Mazur形式群的高度与E mod p的超奇异性之间的关系.对于Calabi-Yau簇上的微分形式，给出了Illusie层的上同调群配对的一个结果.我们还证明了2d次极化K3曲面模空间M_{2d}所包含的完备子簇的最大维数等于17。