Local and canonical heights of subvarieties

Gubler, Walter

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 711-760.

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 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.

Classification : 14G40, 11G50, 14G22

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