ScalingStacks

We now indicate how NA geometry is unified with complex geometry. Consider an algebraic degeneration $X$ over a punctured curve. Let $|\cdot|$ denote the usual absolute value for complex numbers. Given a $\mathbb{C}$-point $z\in X_{t}$ for $0<|t|\ll 1$, inside some affine chart $U=\text{Spec}(A)$ of $X$, we can define a multiplicative seminorm $A\to\mathbb{R}_{\geq 0}$ (not non-archimedean!)

$$f\mapsto e^{-\log|f(z)|/\log|t|}=|f(z)|^{1/|\log|t||}.$$

(13)

As a sequence of points $z$ move towards $t\to 0$, for any given meromorphic function $f=\sum a_{k}t^{k}$ on the base, $\lim_{t\to 0}\log|f(z)|/\log|t|=ord_{0}(f)$ which is the standard NA valuation on $K$. Thus the points on $X_{K}^{an}$ are natural limits of the multiplicative seminorms defined by $\mathbb{C}$-points on $X_{t}$. One can formalize this notion by introducing a hybrid topology on $X\sqcup X_{K}^{an}$, so that $X_{K}^{an}$ takes the place of the central fibre [3, Appendix]. The functions $f\in A$ then induce local continuous functions on $X\sqcup X_{K}^{an}$.