## 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.

### MSC:

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