Corollary 20.4.30.74858EC68B34 Choose decompositions $S_ Z=B_ Z+\mathrm{dd}^{\mathrm{c}}v_ Z$ and $S_{Z'}=B_{Z'}+\mathrm{dd}^{\mathrm{c}}v_{Z'}$ with smooth closed forms $B_ Z,B_{Z'}$, as in corollary 5.38.6. Let $\rho _{E_ g}$ be a global logarithmic potential of $E_ g$, that is, a quasi-plurisubharmonic function on $Z'$ such that $\mathrm{dd}^{\mathrm{c}}\rho _{E_ g}-[E_ g]$ is smooth. Then
\[ v_{Z'}-(v_ Z\circ g)-\rho _{E_ g} \]
is represented by a smooth function. In particular,
\[ v_{Z'}\preceq v_ Z\circ g \]
in the sense of definition 16.7.1, and the two ordinary singularity types are equal if and only if $E_ g=0$.
Proof.
Global logarithmic potentials exist: choose smooth Hermitian metrics on the line bundles of the prime components of $E_ g$, let $\alpha _{E_ g}$ be the corresponding weighted sum of their curvature forms, and let $\rho _{E_ g}$ be the weighted sum of the logarithms of the squared norms of the canonical sections. Then
\[ \mathrm{dd}^{\mathrm{c}}\rho _{E_ g}=[E_ g]-\alpha _{E_ g}, \]
and any two global logarithmic potentials differ by a smooth function, by lemma 5.38.4. We may therefore use this choice. The function $\rho _{E_ g}$ is quasi-plurisubharmonic and bounded above on the compact manifold $Z'$.
The chosen decompositions and (20.4.27) give
\[ \mathrm{dd}^{\mathrm{c}}\bigl (v_{Z'}-v_ Z\circ g-\rho _{E_ g}\bigr ) =g^*B_ Z+\alpha _{E_ g}-B_{Z'}, \]
which is smooth. By lemma 5.38.4, this difference of locally integrable functions is represented by a smooth function. The resulting equality of quasi-plurisubharmonic functions extends from almost everywhere to everywhere by lemma 5.24.3. Since $\rho _{E_ g}$ is bounded above, this gives the ordinary domination.
If $E_ g=0$, the smooth difference gives ordinary equivalence. Conversely, ordinary equivalence would give equal generic Lelong numbers along every prime divisor of $Z'$, whereas the identity just proved shows that along a prime component of $E_ g$ these numbers differ by its positive coefficient. Hence ordinary equivalence forces $E_ g=0$.
$\square$
Pome
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