Theorem 9.11.60E80960D59FB4

Proof. The coefficient below differs from that in (10.23.2) because here $\beta _0=dA/V$. Define on the punctured coordinate disk

Direct differentiation gives

off $q$. The flux of $P_\gamma $ through the circle of radius $\mathrm{e}^{-t}$, which computes the mass of $\mathrm{dd}^{\mathrm{c}}P_\gamma $ inside this circle, is

as $t\to \infty $. Thus the distributional extension across $q$ adds no atom. Also $P_\gamma $ is locally integrable and $|P_\gamma (t)|/t\to 0$, so its Lelong number at $q$ is zero. These are the same flux and growth mechanisms as in theorem 10.23.1, with the present normalization of area.

Take a smooth cutoff $\chi $ supported in the coordinate disk and equal to one near $q$, and extend $L=c_\gamma \chi P_\gamma $ by zero. The signed measure

has a smooth density $\rho $ on all of $E$: near $q$ the singular densities cancel, leaving $-M\beta _0$. Its integral is zero by Stokes' theorem and $c_\gamma F_\gamma =M$. Hence lemma 9.11.51 gives a smooth correction $R=G*(\rho \beta _0)$. Set $v_\gamma =L+R$ and $U_\gamma =\Psi +v_\gamma $. Then

As before, the positive local Laplacians give subharmonic representatives. The disk-lifting argument in the proof of theorem 9.11.54 descends $U_\gamma -\sup _ EU_\gamma $. It is smooth away from the original poles and from $q$. Its non-pluripolar product removes the original atoms, and a possible new value $-\infty $ at $q$ carries no atom. The mass is therefore $M$, and (9.11.25) gives the equation on $X$.

For $\gamma >2$, $P_\gamma \to 0$, so $v_\gamma $ extends continuously and boundedly, and the solution belongs to the prescribed ordinary type. The difference of $v_\gamma $ from its value at $q$ has a nonzero leading term proportional to $t^{2-\gamma }$, with a smooth remainder of order $O(r)$. This decays more slowly than every power $r^\alpha $, $\alpha >0$, so the residual is not Hölder continuous for any positive exponent. Uniqueness follows from the mass of the whole type, the restoration of the fixed pole atoms and the harmonic-difference argument used in the proof of theorem 9.11.54; none of these steps requires $p>1$ once the bounded correction exists.

For $1<\gamma \leq 2$, $P_\gamma \to -\infty $. The correction is bounded above but unbounded below near $q$, so the inequality $\Psi \leq U_\gamma +C$ fails off the pole set. Since its Lelong number is zero, the point atoms of $\beta +\mathrm{dd}^{\mathrm{c}}U_\gamma $ are those of $\beta +\mathrm{dd}^{\mathrm{c}}\Psi $. Applying lemma 5.37.18 to each positive multiple, after a smooth local correction, shows that all multiplier ideals of $U_\gamma $ and $\Psi $ on $E$ coincide, also when $q$ is not one of the original poles. On $X$, this is resolution-ideal equivalence in the sense defined before proposition 9.11.9.

If another solution belonged to the ordinary type, its mass would be $M$, which forces the constant $c_\gamma $. Restoring its prescribed atoms in the extended equation would give exactly (9.11.62). Its difference from $U_\gamma $ would then be harmonic on the torus, hence constant, with equality of the values everywhere. This contradicts the unboundedness of the correction near $q$, so no positive constant admits a solution in the ordinary type.

Finally, the integral of the density is a constant times $\int ^\infty t^{-\gamma }dt$, whereas the integral of its $p$th power, $p>1$, is comparable to

This proves both integrability assertions and completes the proof. $\square$