Generalized heights for arithmetic surfaces
Let \(\pi:{\mathcal X}\to {\mathcal O}_K\) be a regular arithmetic variety of relative dimension \(d\) over the ring of integers in a number field \(K\) and \(\overline{\mathcal E} = ({\mathcal E},h)\) be a hermitian rank \(r\) vector bundle over \({\mathcal X}\) such that \(c_1({\mathcal E}_K)^d > 0\) and \(c_1(\overline{\mathcal E})\) is semi-positive. Following \textit{J.-B. Bost}, the authors define the normalized height of the pair \(({\mathcal X},\overline{\mathcal E})\) by \[ h_{\text{norm}}({\mathcal X},\overline{\mathcal E}): = {1\over [K:{\mathbb{Q}}]} \Biggl[{\widehat c_1(\overline{\mathcal E}^{d+1})\over (d+1)rm}-{\widehat{\deg}\pi_\ast\overline{\mathcal E}\over \text{rk}\pi_\ast{\mathcal E}}\Biggr], \] where \(\widehat c_1(\overline{\mathcal E})\) denotes the arithmetic first Chern class and \(\widehat{\deg}\pi_\ast\overline{\mathcal E}\) the arithmetic degree. If \(X\) is a smooth projective variety of dimension \(d\) over \(\overline{\mathbb{Q}}\) and \(E\) is a rank \(r\) vector bundle over \(X\) generated by its global sections and satisfying \(c_1(E)^d > 0\), one defines \(h_{\text{norm}}(X,E)\in {\mathbb{R}}\cup\{\infty\}\) as \(\inf_{(K,{\mathcal X},\overline{\mathcal E})}h_{\text{norm}}({\mathcal X},\overline{\mathcal E})\), where \((K,{\mathcal X},\overline{\mathcal E})\) is as above with \((X_K,{\mathcal E}_K) \cong (X,E)\). It was shown by Bost that \(h_{\text{norm}}(X,E)\) is bounded from below if \(r = 1\) and the image of \(X\) under the map defined by the complete linear system \(|L|\) is semi-stable in the sense of geometric invariant theory. The authors prove that \(h_{\text{norm}}(X,E) \geq 0\) if \(X\) is a curve and \(E\) is a semi-stable vector bundle with \(h^0(X,E) \geq {\deg(E)\over 2}+r\). They also give a function field analogue of this result.
scientific article; zbMATH DE number 937061