On the height of Calabi-Yau varieties in positive characteristic

Abstract

We study invariants of Calabi-Yau varieties in posi- tive characteristic, especially the height of the Artin-Mazur formal group. We illustrate these results by Calabi-Yau varieties of Fermat and Kummer type. The large measure of attention that complex Calabi-Yau varieties drew in re- cent years stands in marked contrast to the limited attention for their coun- terparts in positive characteristic. Nevertheless, we think these varieties de- serve a greater interest, especially since the special nature of these varieties lends itself well for excursions into the largely unexplored territory of varieties in positive characteristic. In this paper we mean by a Calabi-Yau variety a smooth complete variety of dimension n over a field with dimH i (X,OX) = 0 for i = 1,...,ni1 and with trivial canonical bundle. We study some invariants of Calabi-Yau varieties in characteristic p > 0, especially the height h of the Artin-Mazur formal group for which we prove the estimate h · h 1,ni1 + 1 if h 6 1. We show how this invariant is related to the cohomology of sheaves of closed forms. It is well-known that K3 surfaces do not possess non-zero global 1-forms. The analogous statement about the existence of global i-forms with i = 1 and i = n i 1 on a n-dimensional Calabi-Yau variety is not known and might well be false in positive characteristic. We show that for a Calabi-Yau variety of dimension ¸ 3 over an algebraically closed field k of characteristic p > 0 with no non-zero global 1-forms there is no p-torsion in the Picard variety and Pic/pPic is isomorphic to NS/pNS with NS the Neron-Severi group of X. If in addition X does not have a non-zero global 2-form then NS/pNS ­Fp k maps injectively into H 1 (X,­ 1). This yields the estimate ½ · h 1,1 for the Picard