Arithmetic of Algebraic Varieties and their Moduli Spaces

设X是代数闭域k上n维的非奇异完备代数簇，如果标准丛平凡且结构束的上同调群除0次和n次外零，则称X为Calabi-Yau簇.本文利用Calabi-Yau簇的Artin-Mazur形式群研究了Calabi-Yau簇的结构.作为与范德格尔的联合工作，我得到了以下结果。设X^{r}(P)是特征p&gt；0中维r的Fermat型的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，且h等于1当且仅当p等于1模r+2。Artin提出了一个猜想：特征p&gt；0中的K3曲面X是Shioda意义下的超奇异当且仅当它是Artin意义下的超奇异的。很容易证明，对于Fermat K3曲面X^{2}(P)，猜想成立。使用我们的结果，我们看到我们不能将猜想推广到高维的情况。我们还研究了刚性广义Kummer Calabi-Yau簇X。X的中间Jacobian簇同构于一条椭圆曲线E。我们考虑了约化模p，在某些特殊情况下，我们明确了X mod p的Artin-Mazur形式群的高度与E mod p的超奇性之间的关系。对于Calabi-Yau簇上的微分形式，我们给出了Illusie层的上同调群的配对结果。我们还证明了包含在2d次极化K3曲面的模空间M2d}中的完备子簇的最大维度等于17。

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.