Differentiability of the arithmetic volume function along the base conditions

TL;DR

This paper proves the differentiability of the arithmetic volume function at big pairs of adelic R-Cartier divisors with base conditions, linking its derivative to an arithmetic intersection number.

Contribution

It establishes the differentiability of the arithmetic volume function along base conditions and explicitly characterizes its derivative.

Findings

Arithmetic volume function is differentiable at big pairs.

Derivative is given by an arithmetic restricted positive intersection number.

Provides a new tool for studying the geometry of adelic divisors.

Abstract

In this paper, we show that the arithmetic volume function defined on the space of pairs of adelic R-Cartier divisors and base conditions is differentiable at a big pair, and that its derivative is given by an arithmetic restricted positive intersection number defined for the pair.