9.11.50 Logarithmic equations on normalized curves7AADE341B54C

Proof. With the normalization of $\mathrm{dd}^{\mathrm{c}}$ fixed in the Conventions, $\mathrm{dd}^{\mathrm{c}}v=(\Delta v)/(4\pi )\, dA$ and $\mathrm{dd}^{\mathrm{c}}\log |z|^2=\delta _0$, as in lemma 5.37.18. Cutting off this logarithm in a coordinate disk, with a cutoff equal to one near zero, gives a function $L$ on $E$ with $\mathrm{dd}^{\mathrm{c}}L=\delta _0+q\beta _0$, where $q$ is smooth and $\int _ Eq\beta _0=-1$ by Stokes' theorem.

Let $e_\xi $ be the characters of the torus, with $-\Delta e_\xi =\lambda _\xi e_\xi $. The nonconstant characters have $\lambda _\xi >0$, growing quadratically on the dual lattice. Define the correction $S$, of mean zero, by its Fourier coefficients

where $(q+1)_\xi $ is the Fourier coefficient of $q+1$. Integration by parts shows that these coefficients decay faster than any power of $\lambda _\xi $. Hence the series and all its derivatives converge uniformly, and $\mathrm{dd}^{\mathrm{c}}S=-(q+1)\beta _0$. Conjugate symmetry of the coefficients makes $S$ real. Subtracting the mean of $L+S$ from $L+S$ proves (9.11.52). In particular, $G$ belongs to $L^ r$ for every finite $r$.

The convolution uses the group law and the Haar probability $\beta _0$. Write $K=G*(r\beta _0)$. Testing against smooth functions and using Fubini's theorem proves its distributional equation. Smooth input gives smooth output by the same Fourier argument. For $p'=p/(p-1)$, Hölder's inequality gives

Approximating $G$ by smooth functions shows that translation is continuous in $L^{p'}$, and hence that $K$ is continuous. $\square$

Proof. Set $c=M/F$ and

The lemma gives a bounded continuous $v$ and the positive current

In each disk, adding a smooth local primitive of $\beta $ gives a function with positive distributional Laplacian, and hence a subharmonic representative. Continuity off the poles and the explicit logarithmic values identify this representative with $U$ as defined above. The same argument applies to $\Psi $. The supremum $S=\sup _ EU$ is finite, and we put $u=(U-S)\circ \pi ^{-1}$.

Every holomorphic disk $g:\Delta \to X$ lifts holomorphically to $\widetilde g=\pi ^{-1}\circ g$. Indeed, for a nonconstant disk the inverse image of the finite singular set is discrete, and off this set the lift is holomorphic. Since $\pi ^{-1}$ is a homeomorphism, $\widetilde g$ is continuous on the whole disk. Near an omitted point its image therefore stays in a bounded coordinate neighborhood in $E$, and the removable singularity theorem in one variable extends it holomorphically. A constant disk lifts to a constant map, and a nonconstant disk cannot have image in a finite set.

Choose a smooth local primitive $\eta $ of $\beta $ near the lifted image, that is, $\mathrm{dd}^{\mathrm{c}}\eta =\beta $. Then $\eta \circ \widetilde g$ is a primitive of $g^*h$, and

is subharmonic or identically $-\infty $. That the composition is subharmonic follows from decreasing smooth local regularizations and the holomorphic chain rule for the Laplacian. Any other primitive on the disk differs by a harmonic function. These are exactly the disk tests of definition 9.11.2, and the same argument shows that $\psi \in \mathrm{PSH}(X,h)$. A constant disk at a pole has the permitted restriction $\equiv -\infty $. A nonconstant lift meeting $q_ j$ with order $\ell $ gives the pole coefficient $\ell a_ j$.

The homeomorphism $\pi $ gives continuity off the poles. Near a pole, the smooth logarithmic remainder of $\Psi $ and the bounded continuous function $v$ give continuity with extended values. The residual is the descent of $v-S$, including at the poles, so no difference of two infinite values is involved. The solution belongs to the stated ordinary type.

For the mass and uniqueness assertions, let $w$ be any member of the type and put $W=w\circ \pi $. Pulling back along coordinate disks of the normalization shows that $W$ is $\beta $-plurisubharmonic and not identically $-\infty $. The two type inequalities make $W$ locally bounded away from $Q=\{ q_ j\} $, and at each pole they give

with value $-\infty $ at $q_ j$. The positive Radon measure $T_ W=\beta +\mathrm{dd}^{\mathrm{c}}W$ has total mass $H$ by Stokes' theorem. Its atom at $q_ j$ has weight exactly $a_ j$. This follows from the planar atom identity of lemma 5.37.18. Explicitly, choose a smooth local primitive $\eta $ of $\beta $ near $q_ j$, and let $m(r)$ be the circular mean of $W+\eta $ around $q_ j$. Since $\mathrm{dd}^{\mathrm{c}}(W+\eta )=T_ W$, it satisfies

for almost every $r$, by the divergence theorem and regularization. If $\tau =T_ W(\{ q_ j\} )$, integration gives $m(r)/\log r^2\to \tau $, while the bounded logarithmic remainder gives the limit $a_ j$.

Hence the non-pluripolar product of $W$, defined in dimension one by truncation, is

Indeed, on each relatively compact set avoiding $Q$, a sufficiently low truncation equals $W$, and the indicator removes the poles. Its mass is $H-A=M$ for every member of the type. For (9.11.55), removing the atoms leaves $cf\beta _0$. The definition (9.11.25) gives the asserted equation on $X$. A locally pluripolar set pulls back to a set on which some quasi-subharmonic function, not identically $-\infty $, equals $-\infty $; such sets have zero flat area. Hence $\nu $ is non-pluripolar.

Suppose that $w$ solves the equation with a constant $c'>0$. Comparing total masses gives $c'F=M$, so $c'=c$. Since $\pi $ is a Borel homeomorphism and the atoms are fixed, $T_ W=T_ U$. Thus $\mathrm{dd}^{\mathrm{c}}(W-U)=0$, where $W-U$ is an integrable function. Every nonconstant Fourier coefficient of $W-U$ vanishes, so the difference is constant almost everywhere. Subharmonic functions are determined by their averages over shrinking disks, so the values of $W$ and $U$ differ by the same constant everywhere, and the same holds on $X$. This includes members of the type with no regularity assumed.

Finally, a resolution $\rho :Y\to X$ factors through $E$ by the disk-lifting argument. The resulting proper holomorphic map $Y\to E$ has generic degree one. For smooth compact curves, the multiplicities in a fiber sum to the degree, so every fiber is a single point of multiplicity one, and the map is biholomorphic. The same holds for every common refinement, which proves the independence assertion directly. $\square$