Intrinsic heights of stable varieties and abelian varieties
type: Item
instance of: scholarly article
title: Intrinsic heights of stable varieties and abelian varieties
zbMATH Open document ID: 0867.14010
DOI: 10.1215/S0012-7094-96-08202-2
author: Jean-Benoît Bost
published in: Duke Mathematical Journal
publication date: 1996-09-05T00:00:00Z
review text: This paper, a sequel to: \textit{J.-B. Bost}, ``Semistability and heights of cycles'', Invent. Math. 118, No. 2, 223-253 (1994), defines a ``normalized height'' for a pair consisting of a smooth projective variety \(X\) defined over some number field and a line bundle \(L\) on \(X\) such that \(L\) is generated by its global sections and \(\deg_L X>0\). One first defines the height of a pair consisting of an arithmetic variety \(\mathcal X\) and a hermitian line bundle \(\mathcal L\) on \(\mathcal X\). The normalized height is then the infimum over all models \({\mathcal X}\) of \(X\) satisfying a semipositivity condition relative to the base. The main theorem is: If \(L\) is very ample and maps \(X\) to a Chow stable cycle of degree \(\delta\) and dimension \(d\), then the normalized height of \((X,L)\) is bounded below by an effectively computable constant depending only on \(d\) and \(\delta\). For polarized abelian varieties, the normalized height is bounded above by half the Faltings height. The proof depends on arithmetic Riemann-Roch and ideas of \textit{L. Moret-Bailly} [``Pinceaux de variétés abéliennes'', Astérisque 129 (1985; Zbl 0595.14032)]. From this inequality, one retrieves the important fact that the Faltings height is indeed a height function.
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 11G35
zbMATH DE Number: 908680
zbMATH Keywords: height of a pair
zbMATH Keywords: normalized height
zbMATH Keywords: arithmetic Riemann-Roch
MaRDI profile type: Publication
cites work: Q4098875
cites work: Determinant fiber bundles, Quillen metric and degeneracy of curves.
cites work: Analytic torsion and holomorphic determinant bundles. I: Bott-Chern forms and analytic torsion. II: Direct images and Bott-Chern forms. III: Quillen metrics on holomorphic determinants
cites work: The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle
cites work: Q4010656
cites work: Semi-stability and heights of cycles
cites work: Heights of Projective Varieties and Positive Green Forms
cites work: Q4016341
cites work: Q3699823
cites work: Divisor classes associated to families of stable varieties, with applications to the moduli space of curves
cites work: Zur algebraischen Geometrie. IX. Über zugeordnete Formen und algebraische Systeme von algebraischen Mannigfaltigkeiten
cites work: Q3718844
cites work: Q5190277
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: Calculus on arithmetic surfaces
cites work: Diophantine approximation on abelian varieties
cites work: Lectures on the Arithmetic Riemann-Roch Theorem. (AM-127)
cites work: Q3997943
cites work: Q3328651
cites work: Global moduli for surfaces of general type
cites work: Q3758019
cites work: Arithmetic intersection theory
cites work: Characteristic classes for algebraic vector bundles with Hermitian metric. I
cites work: Characteristic classes for algebraic vector bundles with Hermitian metric. II
cites work: An arithmetic Riemann-Roch theorem
cites work: Q3831122
cites work: Q4195061
cites work: Q4741460
cites work: Instability in invariant theory
cites work: Two Fundamental Theorems on Deformations of Polarized Varieties
cites work: Q3726258
cites work: Q5190276
cites work: Q3351520
cites work: Q5628266
cites work: Stability of projective varieties
cites work: Q4740026
cites work: Sur des hauteurs alternatives. I. (On alternative heights. I)
cites work: Faisceaux amples sur les schémas en groupes et les espaces homogènes
cites work: Q5523386
cites work: Q4010525
cites work: Q4294448
cites work: Q4002219
cites work: Calabi's conjecture and some new results in algebraic geometry
cites work: Positive Line Bundles on Arithmetic Varieties
cites work: Q4763959
full work available at URL: s0012-7094-96-08202-2
OpenAlex ID: W2004696207
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description: scientific article; zbMATH DE number 908680