The spaces $\mathcal H_ m(u)$ need not be closed under multiplication: the product of two sections with integrable weighted squared norms need not have an integrable weighted squared norm, as the following logarithmic example shows. Requiring the weighted pointwise norm to be bounded rather than integrable gives spaces that do multiply. They form the graded algebra used in section 19.9.
Example 19.8.9.D1A2270A67A3 Let $X=\mathbb P^1$ with affine coordinate $z$, and let $\delta _0$ be the Dirac mass at the origin. Take $L=\mathcal O(1)$ with its Fubini–Study metric, whose curvature form is $\beta $, and let
\[ u(z)=\frac35\log \frac{|z|^2}{1+|z|^2}. \]
Then $\beta +\mathrm{dd}^{\mathrm{c}}u=\frac25\beta +\frac35\delta _0\geq 0$, $\mathcal I(u)=\mathcal O$ and $\mathcal I(2u)=\mathcal O(-[0])$. A section of $L$ that does not vanish at $0$ belongs to $\mathcal H_1(u)$, but its square does not belong to $\mathcal H_2(u)$.
Proof.
The curvature formula follows from the Poincaré–Lelong formula $\mathrm{dd}^{\mathrm{c}}\log |z|^2=\delta _0$ and from $\beta =\mathrm{dd}^{\mathrm{c}}\log (1+|z|^2)$ on $\mathbb C$. The weight is bounded near infinity and equals $\frac35\log |z|^2+O(1)$ near zero, its only singularity. The one-variable calculation of example 5.33.3 therefore gives the two stated ideals: $\mathcal I(u)$ is trivial, while $\mathcal I(2u)$ consists of the functions vanishing at the origin. A section of $L$ with $s(0)\neq 0$ thus lies in $\mathcal H_1(u)$, whereas $s^2$ does not vanish at the origin and so does not lie in $\mathcal H_2(u)$.
$\square$
We return to the general data $(X,L,h,u)$. Define $\mathcal W_0=\mathbb C$ and, for $m\geq 1$,
\[ \mathcal W_ m(u)=\{ s\in H^0(X,L^ m): |s|_{h^ m}^2\mathrm{e}^{-mu}\text{ is bounded off }\{ u=-\infty \} \} . \]
These are vector spaces, since $|s+t|^2\leq 2|s|^2+2|t|^2$; they are finite-dimensional and do not depend on the smooth metric, because the ratio of two smooth metrics is bounded above and below on $X$. Since $X$ has finite volume, they are contained in $\mathcal H_ m(u)$. For positive integers $m,k$, $s\in \mathcal W_ m(u)$ and $t\in \mathcal W_ k(u)$, the identity
\[ |st|_{h^{m+k}}^2\mathrm{e}^{-(m+k)u} =\bigl (|s|_{h^ m}^2\mathrm{e}^{-mu}\bigr ) \bigl (|t|_{h^ k}^2\mathrm{e}^{-ku}\bigr ) \]
proves $\mathcal W_ m(u)\mathcal W_ k(u)\subseteq \mathcal W_{m+k}(u)$. Thus $\bigoplus _ m\mathcal W_ m(u)$ is a graded subalgebra of the section ring $\bigoplus _{m\geq 0}H^0(X,L^ m)$ of $L$.
On a smooth projective variety, for big line bundles with metrics of positive curvature current and positive multiplier-ideal volume, the almost-semigroup construction of 19.4 Algebraic partial Okounkov bodies works with the integrable section spaces themselves.
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