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Proposition 18.28.73AC691D964EDB

Proposition 18.28.73.AC691D964EDB Let $\varphi \in \mathrm{PSH}(X,\theta )$ be a potential with analytic singularities such that $\theta _{\varphi }$ is a Kähler current. Then

\[ \beta ^ k_{v,\varphi , \mu } \rightharpoonup \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n=\theta _{P_{\theta ,K}[\varphi ](v)}^ n\quad \textup{as }k\to \infty . \]

Proof. For $w\in C^0(X)$, let $f_ k,g\colon \mathbb {R}\rightarrow \mathbb {R}$ ($k\geq 1$) be defined as

\[ f_ k(t)\coloneqq \mathcal{L}_{k,\varphi }(N^ k_{v+tw,\mu }),\quad g(t)\coloneqq \mathcal{E}_{\theta ,K}^{\varphi }(v+tw). \]

By corollary 18.28.72, for all $t\in \mathbb {R}$, we have

\[ \lim _{k\to \infty } f_ k(t)= g(t). \]

Here we use that $\mu \in \mathrm{BM}(K,v+tw)$ for all fixed $t$, which follows from $\mu \in \mathrm{BM}(K,v)$ and the boundedness of $tw$ on $K$. Note that $f_ k$ is concave by Hölder's inequality. For each $k$, the function $f_ k$ is differentiable and

\[ f_ k'(0)=\int _ X w\, \beta ^ k_{v,\varphi ,\mu }. \]

This follows by differentiating the Gram matrix of the Hilbert norm $N^ k_{v+tw,\mu }$. The function $g$ is differentiable at $0$ by proposition 18.28.27, with

\[ g'(0)=\int _ K w\, \theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal I}(v)}. \]

Applying 21.11.63 to the convex functions $-f_ k$ and $-g$ gives

\[ \lim _{k\to \infty } f_ k'(0)=g'(0), \]

which is equivalent to $\beta ^ k_{v, \varphi ,\mu } \rightharpoonup \theta _{P_{\theta ,K}[\varphi ](v)}^ n$, by proposition 18.28.27. $\square$

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