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 $\mathcal H_2(u)$.
$\square$
Pome
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