Lemma 5.37.18750629715507

Proof. With the normalization of $\mathrm{dd}^{\mathrm{c}}$ fixed in the conventions, $\mu =\mathrm{dd}^{\mathrm{c}}u$ is a positive Radon measure and $a=\mu (\{ x\} )=\nu (u,x)$. After removal of this atom, the distribution $u-a\log |z-x|^2$ has positive complex Hessian, so it has a subharmonic representative $v$, and $\mathrm{dd}^{\mathrm{c}}v$ has no atom at $x$. Since subharmonic functions that agree almost everywhere agree everywhere,

We claim that, for each $t>0$, the function $\mathrm e^{-tv}$ is integrable on some neighborhood of $x$. Indeed, since $\mathrm{dd}^{\mathrm{c}}v$ has no atom at $x$, there is a disk $D=D(x,R)\Subset \Omega $ with $M=(\mathrm{dd}^{\mathrm{c}}v)(D)<1/t$. On $D$ write

where $h$ is harmonic: the integral is locally integrable by Fubini's theorem and has complex Hessian $\mathrm{dd}^{\mathrm{c}}v$ on $D$, so it differs from $v$ by a harmonic distribution. If $M=0$, the claim follows from harmonicity. For $M>0$, Jensen's inequality for the probability measure $(\mathrm{dd}^{\mathrm{c}}v)/M$ gives

Since $tM<1$, the integral of $|z-y|^{-2tM}$ over $z\in D(x,R/2)$ is bounded uniformly in $y\in D$. The harmonic function $h$ is bounded below on this smaller disk, so Tonelli's theorem proves the claim. (Truncating the logarithmic kernel justifies Jensen's inequality at its poles.)

A nonzero holomorphic germ of order $k$ at $x$ is a unit times $(z-x)^ k$. Since $v$ is locally bounded above, weighted integrability requires $k+1>a$; equality is excluded by the logarithmic divergence of the radial integral. Conversely, if $k+1>a$, choose $b>1$ such that $b(a-k)<1$. Then $|z-x|^{2k-2a}$ belongs locally to $L^ b$, and $\mathrm e^{-v}$ belongs locally to $L^{b/(b-1)}$ by the claim, so Hölder's inequality proves integrability. Thus the germ belongs to $\mathcal I(u)_ x$ exactly when $k\geq \lfloor a\rfloor $.

The points $y$ with $\nu (u,y)\geq 1$ form a locally finite set, since each of them carries an atom of mass at least one of the locally finite measure $\mathrm{dd}^{\mathrm{c}}u$. Near any point, therefore, the ideal sheaf is generated by a single power of a coordinate, which proves the last assertion. $\square$