PomeProposition 18.5.2.6E8985733CBC There are two smooth decompositions of the same positive current for which the corresponding values of $W$ are different.
Proposition 18.5.26E8985733CBC
Proof. Use the Hermitian form $\omega $ and the smooth function $g=u_ t$ of example 11.1.12, with $t\ne 0$ sufficiently small. Both $\omega $ and $\omega '=\omega +\mathrm{dd}^{\mathrm{c}}g$ are Hermitian, and
\[ \omega +\mathrm{dd}^{\mathrm{c}}0=\omega '+\mathrm{dd}^{\mathrm{c}}(-g). \]
For a smooth potential in a Hermitian background, all multiplier ideals are trivial and zero is a competitor for its multiplier-ideal envelope, so that envelope is zero. Consequently
\[ W_\omega (0)=\int _ X\omega ^2,\qquad W_{\omega '}(-g)=\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}g)^2. \]
By the explicit formula (11.1.13), the two integrals differ. $\square$
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