On the product of a curve with a Hermitian manifold, a density depending only on the curve variable reduces the Monge–Ampère equation to a Poisson equation on the curve, even when the second factor is not Kähler. Explicit primitives then allow a comparison with the radial families. For a single logarithmic factor, boundedness of the potential and finiteness of the distance have the same threshold, whereas the iterated logarithmic family of theorem 10.22.7 separates them.
Theorem 10.23.1.3BB4C60DE3DF Let $C$ be a compact Riemann surface with a positive $(1,1)$-form $\omega _ C$, let $(N,\omega _ N)$ be a compact Hermitian manifold of dimension $n-1$, and put $h=\omega _ C+\omega _ N$ on $C\times N$, the two forms being pulled back by the projections. Let $p_0\in C$, and let $F\geq 0$ be an integrable function on $C$, smooth and positive away from $p_0$, with $\int _ CF\omega _ C=\int _ C\omega _ C$. Then there is a function $v$ on $C$ such that $v\circ \operatorname {pr}_ C$ solves the equation $\mathrm{MA}^{\mathrm{ext}}_ h(v\circ \operatorname {pr}_ C)=(F\circ \operatorname {pr}_ C)h^ n$, and the current of this solution has no divisorial component along $\{ p_0\} \times N$.
Let $z$ be a coordinate centered at $p_0$, write $r=|z|$ and $\omega _ C=(\mathrm{i}/2\pi )g(z)\, \mathrm{d} z\wedge \mathrm{d}\bar z$, and call a local potential of $\omega _ C+\mathrm{dd}^{\mathrm{c}}v$ near $p_0$ a total normal potential.
If $Fg=C_0r^{2\tau }$ near $p_0$, with $C_0>0$ and $\tau >-1$, then up to a smooth harmonic term the total normal potential is $C_0r^{2(\tau +1)}/(\tau +1)^2$. The solution is bounded and Hölder continuous, hence has minimal singularities, and the normal distance to $\{ p_0\} \times N$ is of order $r^{\tau +1}$. If $\tau =-a$ with $0<a<1$, the density is locally $L^ p$ exactly when $pa<1$. If $\tau $ is a nonnegative integer, the potential is smooth.
If $Fg=C_0r^{-2}t^{-b}$ near $p_0$, with $t=\log (1/r)$, the mass is finite exactly for $b>1$. The total normal potential, which has no atom at $p_0$, is
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10.23.2
\begin{equation} P_ b(t)=\begin{cases} \displaystyle \frac{4C_0t^{2-b}}{(b-1)(b-2)},& b\ne 2,\\[4pt] -4C_0\log t,& b=2, \end{cases} \end{equation}
up to constants and smooth harmonic terms. It is bounded and continuous, equivalently the solution has minimal singularities, precisely when $b>2$, and it is then not Hölder continuous. The normal end has finite distance precisely when $b>2$. For $1<b\leq 2$ the end is complete, and the potential is unbounded and does not have minimal singularities, but it has zero Lelong number and trivial multiplier ideals. No density in this logarithmic family belongs to $L^ p$ for $p>1$.
If $\omega _ C$ and $\omega _ N$ are Kähler, the unbounded solutions have full mass, and uniqueness in the closed theory excludes a different bounded solution for the same measure. A small strict logarithmic barrier along $\{ p_0\} \times N$ exists also when the solutions are bounded and have minimal singularities.
Proof.
On the curve, the real current $F\omega _ C-\omega _ C$ has integral zero and is therefore $\mathrm{dd}^{\mathrm{c}}v$ for a global distribution $v$, unique up to a constant. Its quasi-subharmonic representative is smooth off $p_0$ by the local Poisson equation. Since $F\omega _ C$ has no atom at $p_0$, neither does the current $\omega _ C+\mathrm{dd}^{\mathrm{c}}v$. Pull $v$ back to $C\times N$. Since every wedge product containing two factors from the curve vanishes, on the regular locus
1DB31DDE4B07
10.23.3
\begin{equation} (h+\mathrm{dd}^{\mathrm{c}}v)^ n=n(F\omega _ C)\wedge \omega _ N^{n-1} =(F\circ \operatorname {pr}_ C)h^ n. \end{equation}
By theorem 13.14.3, the extended product agrees with this smooth product off $\{ p_0\} \times N$ and gives that analytic set zero mass. The right side has the same property and finite mass. This proves the global equation, including the case in which the curve potential has a finite value at $p_0$ but is discontinuous there.
For the power density, put $P_\tau (r)=C_0r^{2(\tau +1)}/(\tau +1)^2$. Direct differentiation gives $\partial _ z\bar\partial _ zP_\tau =C_0r^{2\tau }$, and the radial flux of $P_\tau $ through small circles vanishes as $r\downarrow 0$ when $\tau >-1$, so no Dirac term appears. The difference between the total potential and $P_\tau $ is therefore harmonic on the whole disk and smooth. To realize the model globally, cut off the local potential and solve the remaining smooth mean-zero equation on the curve. The normal line element of the metric is comparable to $r^\tau |\mathrm{d} r|$, which gives a distance of order $r^{\tau +1}$: a lower bound for all paths and an upper bound along a radial path. The $L^ p$ assertion is the convergence of $\int _0^\epsilon r^{1-2pa}\, \mathrm{d} r$.
For the logarithmic density, $P_ b''(t)=4C_0t^{-b}$ and $\partial _ z\bar\partial _ zP_ b=P_ b''/(4r^2)=C_0r^{-2}t^{-b}$. The flux tends to zero for $b>1$, which excludes a logarithmic Dirac mass, and the same cutoff and smooth correction give a compact model. The mass of the density near $p_0$ is a constant multiple of $\int ^\infty t^{-b}\, \mathrm{d} t$, finite exactly for $b>1$, while the integral of its $p$th power, for $p>1$, diverges after the logarithmic change of variables. The primitive has a finite limit exactly for $b>2$, with tail of order $t^{2-b}$, which tends to zero more slowly than every power $r^\alpha $. The normal length is a constant multiple of $\int ^\infty t^{-b/2}\, \mathrm{d} t$, which has the same threshold. The projection to the curve bounds the length of every path from below by this normal integral, and a radial path with fixed point of $N$ attains it.
For $1<b\leq 2$, $|P_ b(t)|/t\to 0$, so the Lelong number is zero. For every weight $k>0$, the integrand $\mathrm{e}^{-kP_ b}$ in polar coordinates is bounded by $\exp (-2t+o(t))$ with respect to $\mathrm{d}t$, which is integrable; hence all multiplier ideals are trivial. These properties do not imply bounded ordinary type. In a Kähler product, the normalized curve integral gives full non-pluripolar mass, and uniqueness of full-mass solutions in the closed theory, discussed in section 25.21, excludes another bounded solution. A bounded competitor for the extended operator coincides globally with its deep truncations, so the same exclusion applies there. Finally, for $D=\{ p_0\} \times N$, a smooth metric on $\mathcal O(D)$ with curvature form $\Theta _ D$ and canonical section $s_ D$, and sufficiently small $\delta >0$, we have $h-\delta \Theta _ D>0$, so $\delta \log |s_ D|^2-C$ is a strict barrier.
$\square$
Pome
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