ScalingStacks

A general (singular) Kähler metric $\omega_{\varphi}$ on $X_{s}$ in the polarisation class $s^{-1}[\Delta]$ is given by a potential $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$. The normalising factor $s^{-1}$ is aimed at extracting nontrivial limits as $s\to\infty$. We can completely analogous define the local potentials:

$$\begin{cases}\varphi_{0}=\varphi+\frac{(n+2)}{2s}\log(\sum_{m^{\prime}\in vertex{(\Delta)}}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle}),\\ \varphi_{m}=\varphi+\frac{(n+2)}{2s}\log(\sum_{m^{\prime}\in vertex{(\Delta)}}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle})-\langle m,\text{Log}_{s}(z)\rangle,\end{cases}$$

(19)

which are by definition psh on respective regions.