Remark 19.3.2353FF6E808B22

Remark 19.3.23.53FF6E808B22 Suppose that $X$ is a normal projective variety. Consider a rank $n$ (surjective) valuation $\nu \colon \mathbb {C}(X)^{\times }\rightarrow \mathbb {Z}^ n$ and a closed positive $(1,1)$-current $T$ on $X$. Then we can always define $\nu (T)\in \mathbb {R}^ n$ as follows: Take a resolution $\pi \colon Y\rightarrow X$ such that there is a smooth flag $Y_{\bullet }$ on $Y$ and $g\in \operatorname{Aut}(\mathbb {Z}^ n_{\mathrm{lex}})$ such that

Then we define

This definition does not depend on the choice of $\pi $, by passing to a common resolution and applying proposition 19.3.14, proposition 19.3.17.