Proposition 10.16.35D137EECD5CC

Proof. Step 1: Kähler forms. Let $\lambda $ be a Kähler form on $Z$ and $q\in \mathrm{PSH}(Z,\lambda )$ with $q\leq 0$. The truncations $q_ j=\max (q,-j)$ have mass $\int _ Z\lambda ^ k$ by theorem 11.1.7, or by Stokes' theorem when $k=1$. Restricting their Monge–Ampère measures to $\{ q>-j\} $ and passing to the increasing limit gives

The smooth current $\lambda $ attains equality.

Step 2: the mass bound. Put $A=C\kappa -\eta $. For a positive current $S$ in $b+t\{ \kappa \} $, the current $S+A$ is positive and belongs to $(C+t)\{ \kappa \} $. Since local non-pluripolar products are positive and multilinear, $\langle S^ k\rangle \leq \langle (S+A)^ k\rangle $, and Step 1 applied to the Kähler form $(C+t)\kappa $ yields (10.16.4). In particular $\mathcal R_ Z(b)$ is finite. The definition involves only the class, so the choice of its smooth representative or of the comparison form does not change the supremum.

Step 3: modifications. Let $\rho :W\to Z$ be a modification between compact Kähler manifolds. By proposition 5.43.18, every positive current in $\rho ^*b$ is the pullback of a unique positive current in $b$. Their non-pluripolar measures correspond on the locus where $\rho $ is an isomorphism, and neither charges the exceptional analytic sets. Hence their total masses agree and

Applying this identity on a common Kähler refinement shows that $\mathcal R(a;Y)$ is independent of the resolution. The same argument compares resolutions of $Y$ and $Y'$, since their maps to $X$ agree on a dense open set and therefore everywhere on a common refinement. $\square$