Local heights of subvarieties over non-archimedean fields. (English) Zbl 0906.14013

Let \(Y\) be a complete algebraic variety over a complete non-Archimedean field \(K\). If the valuation is discrete, then local heights of cycles on \(Y\) are well-known. They are given by intersection numbers in models over the discrete valuation ring. To generalize this to the non-discrete case, we replace the algebraic models by formal models over the valuation ring. First, a proper intersection product of Cartier divisors with cycles on a rigid analytic variety \(X\) is defined. Then we extend this intersection product to admissible formal models. It satisfies the usual properties and corresponds to a normalized intersection product in the algebraic situation. A Cartier divisor on a formal model induces a metrized line bundle on \(X\). The metric is called a formal metric. If \(X\) is quasi-compact and quasi-separated, then limits of roots of formal metrics are characterized as those metrics with a continuous extension to the Berkovich-compactification of \(X\). Using for \(X\) the rigid analytic variety associated to \(Y\), we define local heights of cycles as intersection numbers with Cartier divisors on an admissible formal model of \(X\). The dependence on the models is measured by the metrized line bundles. Finally, it is shown that the local heights satisfy five characteristic properties.


14G20 Local ground fields in algebraic geometry
14C25 Algebraic cycles
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