PomeExample 21.12.36.37CEF2EAED80 In general, in the setup of theorem 21.12.31, there exists $m\in M\cap (P\setminus \Delta (\omega ,\varphi ))$ such that $\chi ^ m\in \mathrm{H}^0(X,L\otimes \mathcal{I}(\varphi ))$.
As a concrete example, let us take $P=[0,1]$. Take $\varphi $ so that $\Delta (\omega ,\varphi )=[0,1/2]$. We claim that $\chi ^1$ is $L^2$-integrable.
It suffices to verify the convergence of (21.12.34). Recall that
\[ \nabla G_0(m')=\log \frac{m'}{1-m'},\quad m'\in [0,1], \]
while
\[ \operatorname{Supp}_{[0,1/2]}(a)= \begin{cases} a/2,& \textup{if }a>0;\\ 0,& \textup{otherwise}. \end{cases} \]
Therefore, (21.12.34) becomes
\[ \int _0^{1/2} \frac{m'}{1-m'}\, \mathrm{d}m'+\int _{1/2}^1 \left(\frac{m'}{1-m'}\right)^{1/2}\, \mathrm{d}m'<\infty . \]
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