Free Basis Consisting of Strictly Small Sections

Let X be a d-dimensional projective arithmetic variety and L an invertible sheaf on X . We fix a C∞-hermitian metric | · | of L. In Arakelov geometry, we frequently ask whether H(X,L) has a free basis consisting of strictly small sections. However, we know few about this problem in general. For example, Zhang [7] proved that H(X,nL) possesses a free basis as above for n ≫ 1 if the following conditions are satisfied: (1) XQ is regular, LQ is ample and L is nef on every fiber of X → Spec(Z). (2) The first Chern form c1(L, | · |) is semipositive. (3) For some positive integer n0, there are strictly small sections s1, . . . , sl of n0L such that {x ∈ XQ | s1(x) = · · · = sl(x) = 0} = ∅. From viewpoint of birational geometry, the ampleness of LQ is rather strong. In this paper, we would like to consider the problem under mild and appropriate assumptions. Let R be a graded subring of ⊕∞ n=0 H (X,nL) over Z. For each n, we assign a norm ‖·‖n to Rn⊗Z R in such a way that ‖s · s ′‖n+n′ ≤ ‖s‖n · ‖s′‖n′ holds for all s ∈ Rn⊗Z R and s′ ∈ Rn′ ⊗Z R. Then