Continuous Extension of Arithmetic Volumes

This paper is the sequel of the paper [4], in which we established the arithmetic volume function of C-hermitian Q-invertible sheaves and proved its continuity. The continuity of the volume function has a lot of applications as treated in [4]. In this paper, we would like to consider its continuous extension over R. CONTENTS Introduction 1 1. A multi-indexed version of the fundamental estimation 5 2. Arithmetic volume function 17 3. Uniform continuity of the arithmetic volume function 21 4. Continuous extension of the arithmetic volume function over R 23 5. Approximation of the arithmetic volume function 27 References 33 INTRODUCTION Let X be a d-dimensional projective arithmetic variety. In [4], for a C-hermitian invertible sheaf L on X , we introduce the arithmetic volume v̂ol(L) defined by v̂ol(L) := lim sup n→∞ log#{s ∈ H(X,nL) | ‖s‖sup ≤ 1} nd/d! . By Chen’s recent work [2], “lim sup” in the above equation can be replaced by “lim”. Moreover, in [4], we construct a positively homogeneous function v̂ol : P̂ic(X)⊗ZQ → R of degree d such that the following diagram is commutative: