Tian's partial $C^{0}$-estimate for Fano manifolds

Let $X$ be a smooth Fano variety, let $L=-K_X$, and fix $t_0\in(0,1]$. Define

$$\mathcal{K}(X,t_0)=\{\omega\in c_1(X): \operatorname{Ric}(\omega)\geq t_0\omega\},$$

where the metrics are Kähler. For a metric potential $\phi$ with curvature form $\omega$, choose an orthonormal basis $\{s_i^{(\phi)}\}_{i=0}^{N_k}$ of $H^0(X,-kK_X)$ for the norm $\|s\|_{k\phi}^2=\int_X|s|^2e^{-k\phi}\omega^n$, and define the Bergman function

$$\rho_\omega^{(k)}(x)=\sum_{i=0}^{N_k}|s_i^{(\phi)}|^2e^{-k\phi}.$$

Tian's partial $C^{0}$-estimate. Given $t_0\in(0,1]$, there exist $k>0$ and $\delta>0$ such that $kL$ is very ample and, for every $\omega\in\mathcal{K}(X,t_0)$,

$$\inf_X\rho_\omega^{(k)}(x)\geq\delta.$$

The estimate supplies the uniform Bergman-kernel lower bound needed to relate Tian's continuity method to algebraic geometry and to obtain $C^0$ control of Kähler potentials. The source further says that $k$ can be chosen arbitrarily large; no resolution status is supplied here.