On the height of Calabi-Yau varieties in positive characteristic
On the height of Calabi-Yau varieties in positive characteristic
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DOI：
10.4171/dm/140
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发表时间：
2003-02
影响因子：
0.9
通讯作者：
G.B.M. van der Geer;Toshiyuki Katsura
中科院分区：
文献类型：
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作者：
G.B.M. van der Geer;Toshiyuki Katsura
研究了Calabi-Yau簇的正特征不变量，特别是Artin-Mazur形式群的高度。我们用Fermat和库默型的Calabi-Yau簇来说明这些结果。近年来，复杂的卡-丘品种引起了人们极大的关注，与其积极特征的对立面受到的有限关注形成了鲜明的对比。尽管如此，我们认为这些变种会引起更大的兴趣，特别是因为这些变种的特殊性质很适合进入积极特征的变种的基本上未被探索的领域。本文中我们称Calabi-Yau簇为域上n维光滑完备簇，其中dimHi（X，OX）= 0，i = 1，. ni 1且具有平凡标准丛。研究了特征p > 0的Calabi-Yau簇的一些不变量，特别是Artin-Mazur形式群的高度h，证明了当h为6 1时，h ·h1，ni 1 + 1的估计.我们展示了这个不变量是如何与封闭形式的层的上同调。众所周知，K3曲面不具有非零整体1-形式。关于在n维Calabi-Yau簇上存在i = 1且i = ni 1的整体i-形式的类似陈述是未知的，并且很可能在正特征上是假的。证明了在特征p > 0的代数闭域k上的维数为n = 3的Calabi-Yau簇不存在非零整体1-形式，Picard簇不存在p-挠，Pic/pPic同构于NS/pNS，NS是X的Neron-Severi群.如果X不具有非零的全局2-形式，则NS/pNS Fpk内射映射到H1（X，1）。这就产生了Picard的估计值½ · h 1，1
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