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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 , 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 -field, we deduce corresponding results for global heights of subvarieties.
@article{ASNSP_2003_5_2_4_711_0,
author = {Gubler, Walter},
title = {Local and canonical heights of subvarieties},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {711--760},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 2},
number = {4},
year = {2003},
mrnumber = {2040641},
zbl = {1170.14303},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_4_711_0/}
}
TY - JOUR AU - Gubler, Walter TI - Local and canonical heights of subvarieties JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2003 SP - 711 EP - 760 VL - 2 IS - 4 PB - Scuola normale superiore UR - http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_4_711_0/ LA - en ID - ASNSP_2003_5_2_4_711_0 ER -
%0 Journal Article %A Gubler, Walter %T Local and canonical heights of subvarieties %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2003 %P 711-760 %V 2 %N 4 %I Scuola normale superiore %U http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_4_711_0/ %G en %F ASNSP_2003_5_2_4_711_0
Gubler, Walter. Local and canonical heights of subvarieties. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 711-760. http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_4_711_0/