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Zbl 1170.14303
Gubler, Walter
Local and canonical heights of subvarieties. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 2, No. 4, 711-760 (2003). ISSN 0391-173X; ISSN 2036-2145/e

Summary: Classical results of Weil, Néron 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-Soulé developed 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 locai 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.

MSC 2000:: *14G40 Arithmetic varieties and schemes
11G50 Heights
14G22 Rigid analytic geometry

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