Chern classes of vector bundles on arithmetic varieties (Q1358973)

Let \(X\) be a regular arithmetic surface over a ring of integers of a number field and \(\overline L=(L,h)\) be a Hermitian line bundle on \(X\). An analogue of the Kodaira-Ramanujam vanishing criterion in Arakelov geometry says that, for any \(e\in H^1(X,L^{-1})\), there is an equality \(\widehat c_1(\overline L)^2\leq A\log|e|+B\) with explicit constants \(A\) and \(B\) [see \textit{C. Soulé}, Invent. Math. 116, No. 1-3, 577-599 (1994; Zbl 0834.14013)]. The authors prove a similar inequality for Chern classes of certain Hermitian vector bundles on higher dimensional arithmetic varieties over \({\mathbb{Z}}\). The proof follows the method of Soulé and also relies on an arithmetic analogue of the Bogomolov-Gieseker stability criterion for vector bundles given by \textit{A. Moriwaki} [Am. J. Math. 117, No. 5, 1325-1347 (1995; Zbl 0854.14013)].