Boundedness of the successive minima on arithmetic varieties

Abstract

In this paper, we study the asymptotic behavior of the successive minima associated with high powers of a Hermitian invertible sheaf on an arithmetic variety. As a consequence, we prove that the arithmetic χ ^ \hat {\chi } -volume function, which is introduced by Yuan, is homogeneous, birationally invariant, and continuous on the arithmetic Picard group. We also obtain the arithmetic Hilbert-Samuel formula for vertically nef Hermitian invertible sheaves.