PomeC21A2A6F9E07Lemma 5.37.1.C21A2A6F9E07 If a quasi-plurisubharmonic function has the form $u=c\log |w|^2+O(1)$ with $c>0$ and locally bounded remainder, then for every $t>0$,
The zeroth power is the full structure sheaf.
Proof. The bounded remainder changes the weight only by bounded positive factors. Expand a nonzero holomorphic germ $g(z,w)$ in homogeneous polynomials in $w$, and let $a$ be the smallest degree whose term does not vanish identically. On smaller product neighborhoods, $|g(z,w)|\leq C|w|^ a$. Hence $a+k>tc$ suffices for weighted integrability, since the corresponding radial integral in $\rho =|w|$ is
Conversely, let $z$ be a point near the origin at which the term of degree $a$ does not vanish identically in $w$. Integration of $|g(z,w)|^2$ over the spheres $|w|=\rho $ gives a positive leading term of order $\rho ^{2a}$, so the radial integral in $w$ diverges when $a+k\leq tc$, logarithmically in the case of equality. Such points $z$ form a nonempty open set in every sufficiently small neighborhood, and Fubini's theorem shows that $g$ is not weighted-integrable near the origin. For $d=0$ this is the radial calculation at a point. Thus $g$ is weighted-integrable exactly when $a\geq \lfloor tc\rfloor -k+1$, that is, when $g$ lies in the power of $\mathcal I_ Y$ given in the statement. $\square$
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