Pome

Lemma 18.8.18FA80F8B6F5A

Lemma 18.8.1.8FA80F8B6F5A For $u,v\in \mathrm{PSH}(X,\omega )$, the sum

D93497D2DEF1 18.8.2
\begin{equation} d_\pi ([u],[v])=\sum _{E\subset Y\text{ prime}}w_ E|\nu _ E(u)-\nu _ E(v)| \end{equation}

is finite and defines a pseudometric on $\mathfrak S(X,\omega )$. Changing $\gamma _\pi $ gives an equivalent pseudometric for the fixed modification $\pi $.

Proof. The current $T=\pi ^*\omega +\mathrm{dd}^{\mathrm{c}}(u\circ \pi )$ is positive. Divisorial removal in lemma 5.38.1 gives, for every finite family of prime divisors,

\[ \sum _ Ew_ E\nu _ E(u)\leq \int _ YT\wedge \gamma _\pi ^{n-1}<\infty . \]

Taking the supremum over finite families shows that the sum converges and that only countably many coefficients are positive. The same bound holds for $v$. Bounded changes of $u$ or $v$ do not affect the coefficients, and the triangle inequality holds term by term. Finally, two Hermitian forms on the compact manifold $Y$ bound one another by fixed positive multiples, and so do the corresponding divisor weights. $\square$

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