We will define some local invariants in $x\in{X^{\rm an}}$. On an open affine neighbourhood $U={\rm Spec}(A)$, the point $x$ is given by a multiplicative seminorm $p$ on $A$ and we often write $|f(x)|:=p(f)$ for $f\in A$. Dividing out the prime ideal $I:=\{f\in A\mid p(f)=0\}$, we get a multiplicative norm on the integral domain $B:=A/I$ which extends to an absolute value $|\phantom{a}|_{x}$ on the quotient field of $B$. The completion of this field is denoted by ${\mathscr{H}}(x)$. It does not depend on the choice of $U$ and it may be also constructed analytically. The absolute value of ${\mathscr{H}}(x)$ is denoted by $|\phantom{a}|$ as it extends the given absolute value on $K$. Note that the completed residue field ${\mathscr{H}}(x)$ of $x$ remains the same if we replace the ambient variety $X$ by the Zariski closure of $x$ in $X$.

Let $s(x)$ be the transcendence degree of the residue field of ${\mathscr{H}}(x)$ over ${\tilde{K}}$. The quotient of the value group of ${\mathscr{H}}(x)$ by $\Gamma$ is a finitely generated abelian group and we denote its ${\mathbb{Q}}$-rank by $t(x)$. Finally, we set $d(x):=s(x)+t(x)$. By [Be90], Proposition 9.1.3, we have $\dim(X)=\dim(V)=\sup_{x\in V}d(x)$ for every open subset $V$ of ${X^{\rm an}}$.