Lemma 5.27.33.0EE345C93AFA Fix the data $n,U,v,\varepsilon ,R,\eta ,\chi _\eta $ of lemma 5.27.29, and write $h_ A$ for the smooth plurisubharmonic function $h$ of that lemma, regarded as a function of $A$. Put $D=\{ 0\} \times U$, and write $[D]$ for the current of integration over this hypersurface. Then, for every $1\leq r\leq n$,
\[ (\mathrm{dd}^{\mathrm{c}}h_ A)^ r\longrightarrow \varepsilon [D]\wedge (\mathrm{dd}^{\mathrm{c}}v)^{r-1} \quad \text{as currents on }\mathbb C\times U,\ A\to \infty . \]
The product on the right is integration of the restriction of the smooth form $(\mathrm{dd}^{\mathrm{c}}v)^{r-1}$ over $D$.
Proof.
Write $S=\mathrm{dd}^{\mathrm{c}}v$, $q=\varepsilon \log (|\zeta |^2/R^2)$ and $T_ A=v-A-q$. The rank-one calculation in the proof of lemma 5.27.29 gives
\[ (\mathrm{dd}^{\mathrm{c}}h_ A)^ r=\chi _\eta '(T_ A)^ rS^ r +r\chi _\eta '(T_ A)^{r-1}\chi _\eta ''(T_ A) \, \mathrm dT_ A\wedge \mathrm d^{\mathrm c}T_ A\wedge S^{r-1} \]
off $D$; since $h_ A$ is smooth and $D$ has measure zero, this formula determines the current. Fix a smooth compactly supported test form. On the projection of its support to $U$, the function $v$ and its derivatives are bounded. Hence the support of $\chi _\eta '(T_ A)$ lies in $\{ |\zeta |\leq s_ A\} $, where $s_ A=C\exp (-A/(2\varepsilon ))$ and $C$ depends only on this compact set and the fixed data. The coefficients involving $v$ and $\chi _\eta $ are bounded at the fixed parameters, so the first summand has pairing $O(s_ A^2)$. In the second summand, the term with two derivatives of $v$ also has pairing $O(s_ A^2)$, and the two cross terms have pairing $O(s_ A)$: the derivative of $q$ in the $\zeta $ direction has size $O(|\zeta |^{-1})$, and its area integral over the disk $\{ |\zeta |\leq s_ A\} $ is $O(s_ A)$.
It remains to integrate the term containing $\mathrm dq\wedge \mathrm d^{\mathrm c}q$. The logarithmic change of the radial variable used for (5.27.30) leaves the factor $\varepsilon $ and the probability weight
\[ r\chi _\eta '(t)^{r-1}\chi _\eta ''(t)\, \mathrm dt, \qquad -\eta \leq t\leq \eta . \]
Only the component of the test form in the $z'$ directions enters this term, and on the shrinking circles its coefficients tend uniformly to their values at $\zeta =0$. Integration in the angle therefore gives, in the limit, exactly the pairing with $\varepsilon [D]\wedge S^{r-1}$. This proves the convergence in every degree.
$\square$
Pome
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