ScalingStacks

For the complex geometric interpretation, we assume as usual $X_{K}$ is the base change of an algebraic degeneration family $X$, with an ample polarisation line bundle $L$. For any given NA Fubini-Study metric (9), we can associate a family of Fubini-Study metrics on $(X_{t},L)$:

$$\left\lVert s\right\rVert_{FS,t}(z)=\frac{|s(z)|}{\{\{\sum_{j}|s_{j}(z,t)|^{2}|t|^{2\log\left\lVert s_{j}\right\rVert_{V}}\}^{1/2m}},\quad\forall z\in X_{t}.$$

(10)

Here $s_{j}$ make sense for finite $t$ because they are selected as finite Laurent polynomials in $t$. By our dictionary, we should consider the limit of $\left\lVert s\right\rVert_{FS,t}^{1/|\log|t||}$ as $t\to 0$. Since

$$0\leq\log(\sum_{j}|s_{j}|^{2}|t|^{2\log{\left\lVert s_{j}\right\rVert_{V}}})^{1/2}-\log\max_{j}|s_{j}||t|^{\log\left\lVert s_{j}\right\rVert_{V}}\leq\log(N+1),$$

in the limit the difference between maximum and square length disappears, so $\left\lVert s\right\rVert_{FS,t}^{1/|\log|t||}$ converges to (9) in the hybrid topology on $X\sqcup X_{K}^{an}$.