Theorem 10.17.62D52D521F2D7

Proof. Changing a divisor metric changes $\ell $ by a smooth function, $w$ by its negative, and $\eta _ v$ by the corresponding $\mathrm{dd}^{\mathrm{c}}$-term. Thus $R_ v$ is unchanged. The same cancellation occurs when a smooth change of the background is accompanied by the opposite change of the potential.

Let $\rho :W\to Z$ be a modification with $W$ Kähler. Pulling back the decomposition, with the total divisor $\rho ^*\Gamma $ including all exceptional coefficients, gives a decomposition on $W$ with bounded remainder $w\circ \rho $, residual current $\rho ^*R_ v$ and nef residual class $\rho ^*b_ v$. Its mass is

For two decompositions on different resolutions, pass to a common Kähler refinement and pull both back in this way. At a generic smooth point of a prime component of the union of the two divisors, the difference of the logarithmic terms is $(a_1-a_2)\log |t|^2+O(1)$, where $t$ is a local defining coordinate of the component and $a_1,a_2$ are its two coefficients. The two potentials have bounded difference, and the resulting pair of inequalities forces $a_1=a_2$. Hence the total divisors coincide, and so do the residual classes in the common background. Finally, adding two decompositions adds both the divisors and the bounded remainders. The residual class of the sum is therefore the sum of the nef residual classes, and (10.17.5) computes its mixed terms. $\square$