The core of the proof consists in showing that

$$V_{K,\omega}(x)=\sup\left\{\frac{1}{N}\log||s||_{Nh}(x)\,/\,N\geq 1,s\in\Gamma(X,L^{N})\text{ and }\sup_{K}||s||_{Nh}\leq 1\right\}.$$

Note that for any of the sections $s$ involved in the supremum, $\varphi:=N^{-1}\log||s||_{Nh}$ belongs to $PSH(X,\omega)$ and satisfies $\varphi\leq 0$ on $K$. Therefore $\varphi\leq V_{K,\omega}$.

Conversely fix $x_{0}\in X$ and $a<V_{K,\omega}(x_{0})$. Fix $\varphi\in PSH(X,\omega)$ such that $\sup_{K}\varphi\leq 0$ and $\varphi(x_{0})>a$. Regularizing $\varphi$ (see Appendix) and translating, we can assume $\varphi\in PSH(X,\omega)\cap{\mathcal{C}}^{\infty}(X)$, $\sup_{K}\varphi<0$ and $\varphi(x_{0})>a$. Fix $\varepsilon>0$. Let $B=B(x_{0},r)$ be a small ball on which $\varphi>a$. We choose $B$ so small that the oscillation of $h$ is smaller than $\varepsilon$ on $B$. Let $\chi$ be a test function with compact support in $B$ and such that $\chi\equiv 1$ in $B(x_{0},r/2)$. We can assume w.l.o.g. that $B\subset{\mathcal{U}}_{\alpha_{0}}$ for some $\alpha_{0}$ but $B\cap{\mathcal{U}}_{\beta}=\emptyset$ for all $\beta\neq\alpha_{0}$. This insures that $\chi$ is a smooth section of $L^{N}$ for all $N\geq 1$.

Let $\psi_{1}$ be a smooth positive metric of $L^{N_{1}}\otimes K_{X}^{*}$ on $X$ (this is possible if $N_{1}$ is chosen large enough since $L$ is positive). Let $\psi_{2}$ be a positive metric of $L^{N_{2}}$ on $X$ which is smooth in $X\setminus\{x_{0}\}$ and with Lelong number $\nu(\psi_{2},x_{0})\geq n=\dim_{\mathbb{C}}X$ (this is again possible if $N_{2}$ is large enough, since $L$ is ample). Observe that $\overline{\partial}\chi$ is a smooth $\overline{\partial}$-closed $(0,1)$-form with values in $L^{N}$ (for all $N\geq 1$). Alternatively it is a smooth $\overline{\partial}$-closed $(n,1)$-form with values in $L^{N}\otimes K_{X}^{*}$. Applying Hörmander’s $L^{2}$-estimates (see e.g. [15], chapter VIII) with weight $\psi_{N}:=(N-N_{1}-N_{2})(\varphi+h)+\psi_{1}+\psi_{2}$, we find a smooth section $f$ of $L^{N}$ such that $\overline{\partial}f=\overline{\partial}\chi$ and

$$\int_{X}|f|^{2}e^{-2(N-N_{1}-N_{2})(\varphi+h)-2\psi_{1}-2\psi_{2}}dV_{\omega}\leq C_{1}\int_{X}|\overline{\partial}\chi|^{2}e^{-2\psi_{N}}dV_{\omega}.$$

Note that $\overline{\partial}\chi$ has support in $B\setminus B(x_{0},r/2)$ where $\psi_{N}$ is smooth so that both integrals are finite. Since $\nu(\psi_{2},x_{0})\geq n$, this forces $f(x_{0})=0$. The second integral is actually bounded from above by $C_{2}e^{-2N(a-\varepsilon)}$, where $C_{2}$ is independent of $N$, since $-\varphi<-a$ on $B$ and the oscillation of $h$ is smaller than $\varepsilon$ on $B$. Therefore $s:=\chi-f\in\Gamma(X,L^{N})$ satisfies $s(x_{0})=1$ and

$$\int_{X}|s|^{2}e^{-2N(\varphi+h)}dV_{\omega}\leq C_{3}e^{-2N(a-\varepsilon)},$$

where $C_{3}$ is independent of $N$. Now $\varphi<0$ in a neighborhood of $K$, so the mean-value inequality applied to the subharmonic functions $|s_{\alpha}|^{2}$ yields for all $x$ in $K$,

	$\displaystyle|s|^{2}e^{-2Nh}(x)$	$\displaystyle\leq$	$\displaystyle C_{\delta}\int_{B(x,\delta)}|s|^{2}(y)e^{-2N[\varphi+h](y)}e^{2N[h(y)-h(x)+\varphi(y)]}d\lambda(y)$
		$\displaystyle\leq$	$\displaystyle C_{4}e^{-2N(a-\varepsilon)}$

if $\delta$ is so small that $|\sup_{B(x,\delta)}\varphi|>0$ is bigger than the oscillation of $h$ on $B(x,\delta)$. Therefore $S:=C_{4}^{-1/2}e^{N(a-\varepsilon)}s\in\Gamma(X,L^{N})$ satisfies $\sup_{K}||S||_{Nh}\leq 1$ and $N^{-1}\log||S||_{Nh}(x_{0})\geq a-\varepsilon-\frac{\log C_{4}}{2N}$. Letting $N\rightarrow+\infty$, $\varepsilon\rightarrow 0$ and $a\rightarrow V_{K,\omega}(x_{0})$ completes the proof of the equality.

To conclude observe that by rescaling one gets

	$\displaystyle-\log T_{\omega}(K)=\sup_{X}V_{K,\omega}$
		$\displaystyle=$	$\displaystyle\!\!\!\!\sup\left\{\frac{1}{N}\sup_{X}\log||S||_{Nh}\,/\,N\geq 1,S\in\Gamma(X,L^{N})\text{ and }\sup_{K}||S||_{Nh}=1\right\}$
		$\displaystyle=$	$\displaystyle\!\!\!\!\sup\left\{-\frac{1}{N}\sup_{K}\log||S||_{Nh}\,/\,N\geq 1,S\in\Gamma(X,L^{N})\text{ and }\sup_{X}||S||_{Nh}=1\right\}$
		$\displaystyle=$	$\displaystyle-\log T_{\omega}^{\prime}(K).$

∎