Let $B$ be a irreducible projective variety over a field $k$, which is regular in codimension 1, and $L$ an ample line bundle on $B$. Set $\mathbb{K}=k(B)$. For a prime divisor $v$ on $B$ and $\alpha\in\mathbb{K}^{\times}$, we denote by ${\operatorname{ord}}_{v}(\alpha)$ the order of $\alpha$ at $v$. Fix a constant $c>1$ and denote by $\mathfrak{M}_{\mathbb{K}}$ the set of prime divisors on $B$. For each $v\in\mathfrak{M}_{\mathbb{K}}$, the corresponding absolute value and weight are defined as

$$|\alpha|_{v}=c^{-{\operatorname{ord}}_{v}(\alpha)},\quad n_{v}=\deg_{L}(v).$$

Then $(\mathbb{K},\mathfrak{M}_{\mathbb{K}})$ is an adelic field. Moreover, $\mathbb{K}$ satisfies the product formula, since the degree of a principal divisor is zero,