Theorem 11.45.972A7418012F7

Proof. We construct uniformly bounded residuals on the curve, then use their equation to determine both the normalized limit and the scalar error.

Uniform construction. The kernel has uniformly integrable logarithmic tails and $C_ G=\sup _ x\int |G(x,z)|\omega _1(z)<\infty $. These facts follow by rotation invariance and the radial integral of $|\log \ell +1|$ on $0<\ell <1$; the Poincaré–Lelong formula and $\int _0^1\log \ell \, \mathrm{d}\ell =-1$ give its curvature and mean. Truncating the kernel shows that convolution of bounded densities is continuous.

Temporarily omit subscripts. On continuous mean-zero functions on the curve define

For all late parameters choose fixed bounds $a\leq a_{\max }$, $d\leq d_{\max }$ and $0<h_{\min }\leq H\leq h_{\max }$. Put $B=4a_{\max }h_{\max }/h_{\min }$ and choose $R_0>C_ G(B+a_{\max })$. On the ball $\| w\| _\infty \leq R_0$, for sufficiently small $\sigma $,

Here $\int \ell ^{\sigma d}\omega _1=(1+\sigma d)^{-1}$. Thus $T_\sigma $ maps the mean-zero ball into itself. Since $\int A(w)\omega _1=a$, its density derivative in direction $h$ is

It follows that $\| T_\sigma (w)-T_\sigma (v)\| _\infty \leq 2\sigma B C_ G\| w-v\| _\infty $. The map is a contraction for all sufficiently small exponents, uniformly over the late parameters. Its fixed point $w_ j$ has mean zero and $\| w_ j\| _\infty \leq R_0$.

Write $Z_ j=Z(w_ j)$ and $A_ j=A(w_ j)$. The fixed-point equation gives $\mathrm{dd}^{\mathrm{c}}w_ j=(A_ j-a_ j)\omega _1$, and the potential

has residual form $A_ j\omega _1+b_ j\kappa $, continuous and semipositive. On coordinate balls, mollification of $w_ j$ converges together with its continuous complex Hessian. The local bounded product, expanded after a common strict smooth correction, therefore equals $N b_ j^ kA_ j\, \mathrm{d}W$. Removing $D$ identifies the singular product with the same measure. The identity

shows that (11.45.11) solves the exponential equation in the ordinary type. By lemma 11.45.4, it is $s_ j$.

Residual and normalized limits. Uniform boundedness and dominated convergence give $Z_ j\to m$ and $A_ j\to A_0$ in $L^1(\omega _1)$, with a common bound on $A_ j$. For uniformly bounded densities $F_ j\to 0$ in $L^1$, one has

First letting $j\to \infty $ and then $M\to \infty $ proves $w_ j\to w_0$ uniformly. Although $A_ j=0$ on $D$ and $A_0>0$ there, the $L^1$ convergence suffices for this argument.

Put $S_ j=\sup _ X(\psi _ j+w_ j)$. A point with $\ell =1$ supplies a uniformly bounded competing value. Since $d_ j\geq d_{\min }>0$, the upper bound $d_{\min }\log \ell +R_0$ tends uniformly to $-\infty $ near $D$, so all maximizing points stay in a fixed compact subset of $X\setminus D$. Uniform convergence there gives $S_ j\to S_0$. Hence $v_ j=\psi _ j+w_ j-S_ j$ has the asserted $L^1$ and local uniform limits. For capacity convergence, the complements of $\{ \psi <-T\} $ are compact subsets of $X\setminus D$. Choose a fixed Hermitian form for which $\psi $ is plurisubharmonic and apply (5.29.4); their omitted sublevels have capacity at most $C/T$. Local uniform convergence, followed by $T\to \infty $, proves convergence in capacity, with the common poles omitted.

The measure formula and uniform convergence of $b_ j$ give total-variation convergence to $N b^ kA_0\, \mathrm{d}W=c_0 f\, \mathrm{d}W$. The same bounded product calculation gives this equation for $v_0$. The residual extensions in proposition 11.45.3 show directly that $v_0\sim \phi _{a,d}$. Choose $C$ with $v_0\geq \phi _{a,d}-C$. For every $t\geq 1$, the potential $q_ t=\phi _{a,d}-tC$ satisfies

which proves membership in the relative envelope class by definition 16.28.61.

Scalar centering. The exact formula is $c_ j=a_ j\mathrm{e}^{\sigma _ j S_ j}/Z_ j$. In the difference quotient

the negative part is bounded by $h_{\max }(d_{\max }|\log \ell |+R_0)$ and the positive part by a constant. Dominated convergence gives a limit $\int H(\psi +w_0)\omega _1=mJ$. Since $m_ j\to m>0$,

The identity $c_ j/c_{0,j}=m_ j\mathrm{e}^{\sigma _ j S_ j}/Z_ j$ proves (11.45.10). Subtracting $\sigma _ j^{-1}\log c_{0,j}$ in (11.45.11) leaves $\psi _ j+w_ j-\sigma _ j^{-1}\log (Z_ j/m_ j)$, proving the centered limit in all three topologies. $\square$