Lemma 21.13.38.FCDBEF4DBD54 Let $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ and assume that $\theta _{\varphi }$ is a Kähler current. Then $\varphi $ admits a quasi-equisingular approximation $(\varphi _ j)_ j$ with the following additional properties:
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$\varphi _ j\in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ and $\theta _{\varphi _ j}$ is a Kähler current for all $j\geq 1$;
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for each $j\geq 1$, there are $q_ j\in \mathbb Z_{>0}$ and toric eigen-sections
\[ s_{j,1},\ldots ,s_{j,a_ j}\in \mathrm{H}^0(X,L^{q_ j}) \]
such that
\[ \varphi _ j=\frac{1}{q_ j}\log \sum _{\ell =1}^{a_ j}|s_{j,\ell }|_{h^{q_ j}}^2+\mathcal O(1). \]
Proof.
Apply Demailly's Bergman-kernel approximation theorem [Theorem 13.21, Dem12] to the singular Hermitian metric $h\exp (-\varphi )$ on $L$. Since $\theta _{\varphi }$ is a Kähler current, the theorem applies to the Hilbert spaces
\[ \mathrm{H}^0\bigl (X,L^ q\otimes \mathcal I(q\varphi )\bigr ) \]
for $q\gg 1$, endowed with the $L^2$-norm induced by $h^ q\exp (-q\varphi )$ and a fixed Kähler volume form. The usual decreasing extraction in Demailly's construction gives a quasi-equisingular approximation; taking $q$ sufficiently large at each step ensures that the approximants still have Kähler Chern currents.
It remains to note that this construction is compatible with the torus action. Indeed, both $h$ and $\varphi $ are $T_ c$-invariant, hence the above Hilbert spaces and their $L^2$-inner products are $T_ c$-invariant. Since $T_ c$ is compact and abelian, each of these finite-dimensional Hilbert spaces admits an orthonormal basis consisting of toric eigen-sections. With such a basis, the corresponding Bergman potential has the displayed form and is $T_ c$-invariant. This proves the lemma.
$\square$
Pome
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