Intrinsic heights of stable varieties and abelian varieties
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.
scientific article; zbMATH DE number 908680
scientific article