Theorem 20.4.26400A4F39FD56

Proof. Step 1: the exceptional divisor. Enumerate the prime divisors on $Z$ with positive trace coefficient. By theorem 20.4.5, the pullback of every finite partial sum of $S_ Z$ is at most $S_{Z'}$, and corollary 20.4.24 identifies the increasing limit of these pullbacks with $g^*S_ Z$. Thus $S_{Z'}-g^*S_ Z$ is a positive closed divisorial current. The two currents have equal coefficients along the strict transform of every prime divisor on $Z$, because these coefficients are computed by the same valuation. Every other prime divisor on $Z'$ is exceptional for $g$, and the exceptional analytic set has only finitely many irreducible components on the compact manifold $Z'$. Hence the difference is the integration current of a finite effective real divisor $E_ g$ supported on its divisorial components, and the coefficients of integration currents determine $E_ g$ uniquely.

Step 2: the cocycle identity. Pullback of positive closed $(1,1)$-currents is functorial, since it is defined by composing local plurisubharmonic potentials. Using (20.4.27) for $h$ and for $g$ gives

and uniqueness gives (20.4.28). Additivity and positive homogeneity follow from the same properties of the trace coefficients and the linearity of pullback.

Step 3: approximation. For fixed $m$, apply (20.4.6) to the finitely many prime divisors in the support of the image ideal on $Z$. After division by $m$, it gives $D_{m,Z'}-g^*D_{m,Z}\geq 0$. The coefficients along strict transforms vanish, so this difference is the integration current of an effective real divisor $E_{m,g}$ supported on the exceptional locus.

Choose also a Gauduchon metric $\gamma $ on $Z$. By theorem 20.4.2, the currents $S_ Z-D_{m,Z}$ and $S_{Z'}-D_{m,Z'}$ are positive, and their trace masses tend to zero. Subtracting the two identities for the exceptional currents gives

where both $S_{Z'}-D_{m,Z'}$ and $g^*(S_ Z-D_{m,Z})$ are positive. By (20.4.23), the total variation of the signed measure $([E_{m,g}]-[E_ g])\wedge (\gamma ')^{r-1}$ is therefore at most

Both terms tend to zero, which proves convergence in total variation. $\square$