Proposition 9.11.14DA5FE7ECD89D

Proof. The function is the restriction of a maximum of two plurisubharmonic functions on $\mathbb C^3$. On the axes parametrized by $(z_1,z_2,z_3)=(\zeta ,0,0)$ and $(0,\zeta ,0)$, its logarithmic coefficients are $\sqrt2$ and $2$. An analytic singularity $c\log \sum _ j|f_ j|^2+O(1)$ restricts to a logarithmic coefficient equal to $c$ times an integer on either axis, provided that it is finite off the origin there. The ratio of the two coefficients would therefore be rational, which it is not.

Pullback by the finite degree-two map reduces local integrability to the cover. Neighborhoods of the origin contain and are contained in images of sufficiently small bidisks, so it suffices to integrate over a bidisk in $s,t$. The pullback weight is

For $s^ at^ b$, $a,b\geq 0$, set $p=-\log |s|^2$ and $q=-\log |t|^2$. By (9.11.11), angular integration gives, up to a positive constant,

for the boundary $D$. For zero boundary, replace $a+\tfrac 12$ by $a+1$. Thus lemma 9.11.13 gives the respective conditions

These monomial tests determine the entire ideal. Indeed, the measure and the weight on the cover are invariant under separate rotations of $s,t$. Parseval's identity and Tonelli's theorem show that every nonzero coefficient of a weighted-integrable holomorphic series must satisfy its monomial test. Only exponents with $a+b$ even occur.

In the first test, the minimal allowed exponents, under addition of nonnegative even-total exponents, are

To see this, every term with $a\geq 2$ is divisible by $s^2$, every term with $a=1$ has odd $b\geq 1$ and is divisible by $st$, and a term with $a=0$ has even $b$ and passes the test exactly when $b\geq 4$. Each of the three displayed exponents passes the test. Splitting a convergent invariant series into these three groups and dividing gives convergent invariant coefficient series on a smaller bidisk. They descend to holomorphic germs on $X$, bounded on a smaller neighborhood. Since the three generators are weighted-integrable, every such combination is weighted-integrable. This proves the first ideal at the origin. For the second test, every nonconstant even-total monomial is allowed, with minimal exponents $(2,0),(1,1),(0,2)$, and the same argument gives the second ideal.

Away from the origin, $u$ is locally bounded and $X$ is smooth. The only additional volume singularity lies along $E$: with $z_2\neq 0$, $z_3$ is a normal coordinate and $z_1=z_3^2/z_2$, so $\mathrm{d}V_ D$ is a smooth positive factor times $|z_3|^{-1}$ times smooth volume. This is locally integrable. Both ideals therefore equal $\mathcal O_ X$ there, as do the ideals generated by the displayed coordinates, which proves the sheaf identities. $\square$