Theorem 11.34.21.D68D3638A657 Let $g$ be a smooth real function on the second factor with $\gamma =\mathrm{dd}^{\mathrm{c}}g\geq -\omega _2$, regarded as a function on $X$. For $0<\delta \leq 1$ put
\[ u_{\delta ,g}=\log (r+\delta \mathrm e^ g),\qquad \Omega _{\delta ,g}=\theta +\mathrm{dd}^{\mathrm{c}}u_{\delta ,g},\qquad F_{\delta ,g}=\log \frac{\Omega _{\delta ,g}^2}{4\mathrm dV}. \]
The forms satisfy $\Omega _{\delta ,g}\geq \omega $ and $\int _ X\Omega _{\delta ,g}^2=8$. For each fixed $g$, as $\delta \downarrow 0$ the smooth potentials $u_{\delta ,g}$ decrease pointwise to $\psi $, converge in $L^1$ and in the capacity $\operatorname {Cap}_\omega $, and converge smoothly off $D$. Moreover $F_{\delta ,g}\to 0$ in $L^ p(\mathrm dV)$ for every $1\leq p<\infty $. The adjoint cofactors and adjoint probabilities of the forms $\Omega _{\delta ,g}$ are
\[ Q_{\delta ,g}=\Omega _{\delta ,g}/8,\qquad \nu _{\delta ,g}=\Omega _{\delta ,g}^2/8, \]
and they have the weak limits
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\begin{equation} \begin{split} Q_{\delta ,g}& \longrightarrow (\beta +[D])/8,\\ \nu _{\delta ,g}& \longrightarrow \tfrac 12\mathrm dV+\tfrac 12[D]\wedge \omega _2 +\tfrac 18[D]\wedge \mathrm{dd}^{\mathrm{c}}g. \end{split} \end{equation}
The limiting probability has mass $1/2$ on $D$ and depends on $g$, although $\mathcal A_\varphi (0)=\{ \mathrm dV\} $. The cofactors do not approach $\beta /4$, and their limit is not $\rho \beta $ for any locally integrable scalar $\rho $.
Proof.
Smoothness and monotone convergence follow directly from the defining formula, also on $D$, where $\psi =-\infty $. Off $D$, set
\[ \lambda =\frac{r}{r+\delta \mathrm e^ g},\qquad h=\psi -g-\log \delta . \]
The chain rule gives $\Omega _{\delta ,g}=A_\delta +R_\delta $, where
\[ A_\delta =(2-\lambda )\omega _1+2\omega _2+(1-\lambda )\gamma , \qquad R_\delta =\lambda (1-\lambda )\mathrm dh\wedge \mathrm d^{\mathrm c}h. \]
Thus $A_\delta \geq \omega _1+\omega _2$ and $R_\delta \geq 0$, and the inequality $\Omega _{\delta ,g}\geq \omega $ extends across $D$ by continuity. Closedness of $\theta $ and Stokes' theorem give the mass eight. Since $\Omega _{\delta ,g}$ is closed, the Gauduchon condition $\mathrm{dd}^{\mathrm{c}}(\rho \Omega _{\delta ,g})=0$ reduces to the elliptic equation $\mathrm{dd}^{\mathrm{c}}\rho \wedge \Omega _{\delta ,g}=0$ on the connected surface, whose smooth solutions are constant by the maximum principle. Normalization against $\Omega _{\delta ,g}^2$ gives $\rho =1/8$ and the stated adjoint cofactors and probabilities.
The rank-one form $R_\delta $ has square zero, hence $\Omega _{\delta ,g}^2=A_\delta ^2+2A_\delta \wedge R_\delta $. The bounded coefficients of $A_\delta $ converge off $D$ to those of $\beta $, which gives $A_\delta ^2\to 4\mathrm dV$. Write $N=\mathrm d\psi \wedge \mathrm d^{\mathrm c}\psi $. In $A_\delta \wedge R_\delta $, the part in which $R_\delta $ contributes only derivatives along the first factor (the normal part) is
\[ \lambda (1-\lambda )N\wedge \bigl (2\omega _2+(1-\lambda )\gamma \bigr ). \]
Terms mixing a derivative along the first factor with one along the second vanish for degree reasons. The remaining tangential term is $\lambda (1-\lambda )(2-\lambda )\omega _1\wedge \mathrm dg\wedge \mathrm d^{\mathrm c}g$, which tends to zero by dominated convergence.
The normal coefficients can be computed in polar coordinates. Put $s=\delta \mathrm e^{g}$ and $t=|z|^2$. Then
\[ \lambda =\frac{t}{s+(1+s)t},\qquad N=\frac{\mathrm i\, \mathrm dz\wedge \mathrm d\bar z}{2\pi t(1+t)^2}. \]
After angular integration the radial measure of $\lambda (1-\lambda )N$ is
\[ \frac{s\, \mathrm dt}{(1+t)(s+(1+s)t)^2} =\frac{\mathrm dy}{(1+sy)(1+(1+s)y)^2},\qquad t=sy. \]
It is bounded by $\mathrm dy/(1+y)^2$ and tends to that measure, whose integral is one. With an additional factor $1-\lambda $, the rescaled measure becomes $\mathrm dy/(1+(1+s)y)^3$, which is bounded by $\mathrm dy/(1+y)^3$ and has limiting integral $1/2$. The mass of both measures outside every fixed tube around $D$ tends to zero, uniformly in the point of the second factor because $g$ is bounded. Testing against continuous functions therefore gives
\[ \lambda (1-\lambda )N\longrightarrow [D],\qquad \lambda (1-\lambda )^2N\longrightarrow \tfrac 12[D]. \]
Combining the normal and the remaining terms proves
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\begin{equation} \Omega _{\delta ,g}^2\longrightarrow 4\mathrm dV+[D]\wedge (4\omega _2+\gamma ). \end{equation}
Dividing by eight gives the second line of (11.34.22). The limit is positive since $4\omega _2+\gamma \geq 3\omega _2$, and its mass on $D$ is $4/8=1/2$ by Stokes' theorem on the second factor. Taking $g=0$ and a small nonconstant smooth $g$ gives different limits, since $\mathrm{dd}^{\mathrm{c}}g$ vanishes identically only for constant $g$.
The difference of the potentials satisfies
\[ 0\leq u_{\delta ,g}-\psi \leq \log \bigl (1+\delta \mathrm e^{\max g}/r\bigr ) \]
off $D$. The right-hand side tends to zero and is dominated by an integrable logarithmic pole, which proves $L^1$ convergence. Continuity of distributional differentiation then gives the limit of the cofactors. A locally integrable function times the smooth form $\beta $ has absolutely continuous coefficients, so it cannot produce the singular coefficients of $[D]$.
For convergence in capacity, use the capacity $\operatorname {Cap}_\omega $ of section 5.29; after subtracting one, its test potentials range over $v\in \mathrm{PSH}(X,\omega )$ with $-1\leq v\leq 0$. For such $v$, apply the cutoff argument of (11.34.20) with $\beta $ replaced by $\omega $ and $w=v+1$. Stokes' theorem gives $\int _ X\omega _1\wedge (2\omega +\mathrm{dd}^{\mathrm{c}}v)=2$, so
\[ \operatorname {Cap}_\omega \{ \psi <-t\} \leq \int _ X(-f_ t)\omega ^2+2/t\longrightarrow 0. \]
The same bound shows that $D$ has capacity zero. For $\eta >0$, the set of points off $D$ where $u_{\delta ,g}-\psi >\eta $ is contained in
\[ \left\{ r<\frac{\delta \mathrm e^{\max g}}{\mathrm e^\eta -1}\right\} . \]
This is a sublevel set of $\psi $ whose level tends to $-\infty $ as $\delta \downarrow 0$. The capacity bound for such sublevel sets, together with $\operatorname {Cap}_\omega (D)=0$, proves convergence in capacity.
Finally, $\Omega _{\delta ,g}\geq \omega $ gives $F_{\delta ,g}\geq -\log 2$. In the affine chart $|\mathrm d\psi |_\omega ^2\leq C/r$, while $g$ and its first two derivatives are bounded. The decomposition above yields $\Omega _{\delta ,g}\leq C_ g(1+r^{-1})\omega $, hence
\[ |F_{\delta ,g}|\leq C_ g+2|\log r|. \]
Every finite power of this logarithmic bound is integrable. Smooth convergence off $D$ gives $F_{\delta ,g}\to 0$ there, and dominated convergence proves convergence in $L^ p$.
$\square$
Pome
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