Subsheaves of a Hermitian Torsion Free Coherent Sheaf on an Arithmetic Variety

Let K be a number field and OK the ring of integers of K. Let (E, h) be a hermitian finitely generated flat OK-module. For an OK-submodule F of E, let us denote by hF →֒E the submetric of F induced by h. It is well known that the set of all saturated OK-submodules F with d̂eg(F, hF →֒E) ≥ c is finite for any real numbers c (for details, see [4, the proof of Proposition 3.5]). In this note, we would like to give its generalization on a projective arithmetic variety. Let X be a normal and projective arithmetic variety. Here we assume that X is an arithmetic surface to avoid several complicated technical definitions on a higher dimensional arithmetic variety. Let us fix a nef and big C-hermitian invertible sheaf H on X as a polarization of X . Then we have the following finiteness of saturated subsheaves with bounded arithmetic degree, which is also a generalization of a partial result [5, Corollary 2.2].