Semi-stability and heights of cycles
type: Item
instance of: scholarly article
title: Semi-stability and heights of cycles
zbMATH Open document ID: 0873.14028
DOI: 10.1007/BF01231533
author: Jean-Benoît Bost
published in: Inventiones Mathematicae
publication date: 1997-05-20T00:00:00Z
full work available at URL: https://eudml.org/doc/144236
review text: Let \(K\) be a number field, \({\mathcal O}_K\) the ring of integers of \(K\), and \(\overline E\) a non-zero hermitian vector bundle over \(S:= \text{Spec} {\mathcal O}_K\). Denote \(\mathbb{P}(E)= \text{Proj(Sym}^\bullet E^\vee)\), and by \(\pi: \mathbb{P}(E)\to S\) the canonical morphism. The canonical quotient line bundle \({\mathcal O}_E(1)\) on \(\mathbb{P}(E)\) will be equipped with the hermitian metric; the hermitian line bundle thus defined will be denoted \(\overline {{\mathcal O}_E (1)}\). To any algebraic cycle \(Z\) of (absolute) dimension \(d\) on \(\mathbb{P}(E)\), we can attach its height with respect to \(\overline {{\mathcal O}_E(1)}\), namely the real number \(h_{\overline {{\mathcal O}_E(1)}} (Z)= \widehat {\deg} \bigl(\widehat c_1 (\overline {{\mathcal O}_E(1)})^d |Z)\). When \(d\geq 1\), we may also introduce the degree of its generic fiber \(\deg_K(Z)= \deg_K (c_1 ({\mathcal O}_E(1))^{d-1} \cdot [Z_K])\). On the other hand, we shall denote by \(\text{rk} E\) the rank of \(E\), and by \(\widehat {\deg} \overline E\) the Arakelov degree of \(\overline E\), that is the real number \(\widehat {\deg} \overline E= \widehat {\deg} \widehat c_1 (\overline E)\). -- Our main result is the following: Theorem I. Let \(Z\) be an effective cycle of dimension \(d\geq 1\) on \(\mathbb{P}(E)\). If the restriction \(Z_K\) of \(Z\) to the generic fiber \(\mathbb{P}(E_K)\) is not zero and has a semi-stable Chow point, then the following inequality holds: \[ {1\over d} {h_{\overline {{\mathcal O}_E (1)}} (Z) \over \deg_K(Z)} \geq- {\widehat {\deg} \overline E\over \text{rk} E} -[K:\mathbb{Q}] C(\text{rk} E,d, \deg_KZ), \] where \(C(\text{rk} E,d, \deg_KZ)\) denotes a constant which depends only on \(\text{rk} E,d\), and \(\deg_KZ\). For various applications, it is convenient to have a more ``intrinsic'' version of theorem I, as follows: Theorem II. Let \(\pi: X\to S=\text{Spec} {\mathcal O}_K\) be any projective arithmetic variety and let \(L\) be an invertible sh…
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 703869
zbMATH Keywords: algebraic cycle
zbMATH Keywords: height
zbMATH Keywords: Arakelov degree
zbMATH Keywords: effective cycle
zbMATH Keywords: semi-stable Chow point
zbMATH Keywords: arithmetic variety
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 703869