Arithmetic of Algebraic Varieties and their Moduli Spaces

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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.