We mentioned in section 3.2 that for a given polarized algebraic degeneration, the choice of snc models is highly non-unique. There are two viewpoints on extracting invariant information:

• •

  In birational geometry, one aims to find optimal representatives via the minimal model program. Typically, this will leave the snc world, and require divisorial log terminal models [62][63], but the minimal models may still be non-unique.

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  In non-archimedean geometry, one looks simultaneously at the tower of all snc models, and take the formal limit of their dual complexes, known as the Berkovich space.

Good references can be found in [49, A] [3, Appendix][6, chapter 2,3].

An insight of Berkovich is that by thinking of points as multiplicative seminorms, one obtains a kind of geometry analogous to complex manifolds. Let $K\simeq\mathbb{C}(\!(t)\!)$ be equipped with its standard absolute value $|\cdot|_{0}=e^{-ord_{t}}$ where $ord_{t}$ is the valuation defined by the vanishing order. Its ultrametric property

gives the name ‘non-archimedean’ to the subject. Let $X_{K}$ be a smooth, geometrically connected, projective scheme over $\text{Spec}(K)$; the main examples come from base changing an algebraic degeneration family $X$ over a punctured curve. Choose a finite cover of $X_{K}$ by affine open sets of the form $U=\text{Spec}(A)$, where $A$ is a finitely generated $K$-algebra. The space $U^{an}$ is defined as the set of all multiplicative seminorms $|\cdot|_{x}:A\to\mathbb{R}_{\geq 0}$ extending the absolute value of $K$, endowed with the weakest topology so that the function $x\mapsto|f|_{x}$ is continuous for any $f\in A$. The Berkovich space $X_{K}^{an}$ is then obtained by gluing together $U^{an}$; the notation stands for ‘analytification’. As a topological space $X_{K}^{an}$ is compact and Hausdorff. In the CY case, the point-set description of $X_{K}^{an}$ is meant to encode information about the base of the SYZ fibration; there is also a natural structure sheaf which encodes information about the complex structure [49].

Let $R\simeq\mathbb{C}[\![t]\!]$. The concept of models over $\text{Spec}(R)$ is entirely analogous to the case over algebraic curves. The dual intersection complexes $\Delta_{\mathcal{X}}$ for snc models over $\text{Spec}(R)$ can be compared with $X_{K}^{an}$ through two natural maps:

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  There is a continuous embedding map $emb:\Delta_{\mathcal{X}}\to X_{K}^{an}$. Writing $\mathcal{X}_{0}=\sum b_{i}E_{i}$, each divisor $E_{i}$ defines $val_{E_{i}}=\frac{ord_{E_{i}}}{b_{i}}$ through the vanishing order $ord_{E_{i}}$, so that $e^{-val_{E_{i}}}$ is a point in $X_{K}^{an}$, called a divisorial point. More generally, given a point $x=(x_{0},\ldots x_{p})$ in the interior of a face $\Delta_{J}\subset\Delta_{\mathcal{X}}$ corresponding to $E_{J}=\cap_{0}^{p}E_{i}$, we can associate a quasi-monomial valuation: expanding any local function $f$ around $E_{J}$ in Taylor series,

  $$f=\sum_{\alpha\in\mathbb{N}^{p+1}}f_{\alpha}z_{0}^{\alpha_{0}}\ldots z_{p}^{\alpha_{p}},\quad f_{\alpha}\in K(E_{J})$$

  then the quasi-monomial valuation is

  $$val_{x}(f)=\min\{\sum_{0}^{p}\alpha_{i}x_{i}|f_{\alpha}\neq 0\}.$$

  Thus $x$ gives rise to a point $e^{-val_{x}}\in X_{K}^{an}$. We shall regard $\Delta_{\mathcal{X}}$ as a subset of $X_{K}^{an}$. In particular, the essential skeleton embeds into $X_{K}^{an}$.

• •

  There is a continuous retraction map $r_{\mathcal{X}}:X_{K}^{an}\to\Delta_{\mathcal{X}}$, which restricts to the identity on $\Delta_{\mathcal{X}}\subset X^{an}$. Any point $e^{-v}\in X_{K}^{an}$ admits a center on $\mathcal{X}$. This is the unique scheme theoretic point $\xi\in X_{0}$ such that $|f|_{x}\leq 1$ for $f\in\mathcal{O}_{\mathcal{X},\xi}$ and $|f|_{x}<1$ for $f\in m_{\mathcal{X},\xi}$. Let $J\subset I$ be the maximal subset such that $\xi\in E_{J}$. Then $r_{\mathcal{X}}(x)\in\Delta_{\mathcal{X}}$ corresponds to the quasi-monomial valuation with the same value for $-\log|z_{j}|_{x},j\in J$. Concretely, one should think the retraction map is about reading off logarithmic coordinates.

  For another perspective, if $\mathcal{X}^{\prime}$ is a blow up of $\mathcal{X}$, then there is a natural simplicial map $\Delta_{\mathcal{X}^{\prime}}\to\Delta_{\mathcal{X}}$, which is identity on $\Delta_{\mathcal{X}}\subset\Delta_{\mathcal{X}^{\prime}}$. The retraction map $X_{K}^{an}\to\Delta_{\mathcal{X}}$ can be viewed as a formal limit for very large $\mathcal{X}^{\prime}$.

With these comparison maps, the Berkovich space $X_{K}^{an}$ is homeomorphic to the inverse limit of the dual intersection complexes of the snc models:

Conceptually, an snc model gives a finite approximation of the Berkovich space.

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!)

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

The ‘hybrid’ space $X\sqcup\Delta_{\mathcal{X}}$ discussed in section 3.1 can be understood as a finite approximation. Given an snc model $\mathcal{X}$, and take a sequence of $\mathbb{C}$-points $q_{k}$ tending to $e^{-v}\in X^{an}$, whose image under the retraction map $r_{\mathcal{X}}$ is $x=(x_{0},\ldots x_{p})\in\Delta_{J}\subset\Delta_{\mathcal{X}}$. Tautologically $q_{k}$ concentrate near $E_{J}$, and in the local coordinates $z_{0},\ldots z_{p}$, we have $\log|z_{i}(q_{k})|/\log|t|\to v(z_{i})=x_{i}$, which is equivalent to $\text{Log}_{\mathcal{X}}(z_{k})\to x=(x_{0},\ldots x_{p})\in\Delta_{J}\subset\Delta_{\mathcal{X}}$. Formally, the topology on $X\sqcup X_{K}^{an}$ is the inverse limit of $X\sqcup\Delta_{\mathcal{X}}$ by taking all snc models.