Local and canonical heights of subvarieties
Local and canonical heights of subvarieties
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亚品种的当地高度和规范高度
DOI：
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发表时间：
2003
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通讯作者：
Walter Gubler
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作者：
Walter Gubler
将Weil， Neron和Tate的经典结果推广到关于厄米特伪因子的子变体的局部高度。如果支撑点的交点为空，则局部高度定义良好。在阿基米德的情况下，度量是厄米的，局部高度是由Gillet-Soule的*产品的改进版本定义的，该产品在紧凑型品种上发展而不假设规则性。在非阿基米德情况下，局部高度是使用刚性几何和形式几何的方法来处理非离散值的交数。为了包含代数等价于0的线束的规范度量，在估值环上的形式模型上引入了局部Chow上同调。利用Tate极限论证，得到了关于任意伪因子的阿贝尔变体上子变体的正则局部高度。通过在m域上的积分，我们推导出了子变体整体高度的相应结果。
Classical results of Weil, Neron and Tate are generalized to local heights of subvarieties with respect to hermitian pseudo-divisors. The local heights are well-defined if the intersection of supports is empty. In the archimedean case, the metrics are hermitian and the local heights are defined by a refined version of the *-product of Gillet-Soule developped on compact varieties without assuming regularity. In the non-archimedean case, the local heights are intersection numbers using methods from rigid and formal geometry to handle non-discrete valuations. To include canonical metrics of line bundles algebraically equivalent to 0, a local Chow cohomology is introduced on formal models over the valuation ring. Using Tate’s limit argument, canonical local heights of subvarieties on an abelian variety are obtained with respect to any pseudo-divisors. By integration over an M-field, we deduce corresponding results for global heights of subvarieties.