20.4.4 Compatibility between models70BE0BB07401

Proof. Every prime divisor on $Z'$ is either the strict transform of a prime divisor on $Z$ or exceptional for $g$. If $E'$ is the strict transform of a prime divisor $E\subset Z$, then $g$ is biholomorphic near generic points of $E'$. Hence $E'$ and $E$ define the same vanishing order on the image ideals from $Y$, so $\tau _{Y,u}(E')=\tau _{Y,u}(E)$, and $g_*[E']=[E]$.

The image of an exceptional prime divisor $F$ has codimension at least two. The pullback of a smooth $(r-1,r-1)$ test form therefore restricts to zero on the regular locus of $F$, where the restriction of $g$ has complex rank at most $r-2$; thus $g_*[F]=0$. In dimension one, a modification between smooth compact curves is an isomorphism, and there are no exceptional divisors. Pushforward under the proper map $g$ is continuous on currents. Applying it to the increasing finite partial sums, or equivalently testing their summable positive coefficient measures, proves the displayed compatibility. Additivity and homogeneity follow coefficientwise from proposition 20.3.4.

For the inequality, fix a positive integer $m$ and let $\mathfrak a_{m,Z}=\mathcal I(mu)\mathcal O_ Z$ denote the image ideal on $Z$. Its divisorial vanishing gives

Indeed, on a manifold, a holomorphic function vanishing to the indicated generic orders is divisible by the corresponding powers of the local equations of the divisors. Pulling back this inclusion and taking the order along a prime divisor $F\subset Z'$ gives

Divide by $m$ and let $m\to \infty $; the sum is finite. Thus every coefficient of $g^*A_{\mathcal F}$ is at most the corresponding coefficient $\tau _{Y,u}(F)$ of $S_{Z'}(u;Y)$. Since both sides are positive divisorial currents, this proves the asserted inequality of currents. $\square$