ScalingStacks

A fundamental result in Kähler geometry is that any Kähler metric in an integral class can be approximated by Fubini-Study metrics associated with projective embeddings. While the usual Fubini-Study metric depends on a choice of a Hermitian inner product on the $\mathbb{C}$-vector space of global sections, the NA analgoue depends on a NA norm on the $K$-vector space $V=H^{0}(X_{K},mL)$ for $m\gg 1$, with the ultrametric property $\left\lVert x+y\right\rVert_{V}\leq\max\{\left\lVert x\right\rVert_{V},\left\lVert y\right\rVert_{V}\}$. In our case $K=\mathbb{C}(\!(t)\!)$ one can select a $K$-basis $s_{0},s_{1},\ldots s_{N}$ for $V$ (called an ‘orthogonal basis’ [17, section 2.1.2]), such that

$$\left\lVert a_{0}s_{0}+\ldots+a_{N}s_{N}\right\rVert_{V}=\max\{|a_{0}|\left\lVert s_{0}\right\rVert_{V},\ldots,|a_{N}|\left\lVert s_{N}\right\rVert_{V}\},\quad\forall a_{i}\in K.$$

The NA Fubini-Study metric on $L\to X_{K}$ can be defined as

$$\left\lVert s\right\rVert_{FS}(x)=\inf_{\tilde{s}\in V,\tilde{s}(x)=s^{\otimes m}(x)}\left\lVert\tilde{s}\right\rVert_{V}^{1/m},\quad\forall x\in X_{K}^{an}.$$

Concretely in the orthogonal basis, written in a local trivialisation,

$$\left\lVert s\right\rVert_{FS}(x)=\frac{|s(x)|}{\max_{j}\{|s_{j}(x)|/\left\lVert s_{j}\right\rVert_{V}\}^{1/m}},\quad\forall x\in X^{an}.$$

(9)

A NA analogue of the Fubini-Study approximation theorem gives an alternative view on semipositive metrics:

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 $X\sqcup X_{K}^{an}$.