Subsection 3.2 (00B6) · ScalingStacks

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3.2 Berkovich space, hybrid topology

One problem with the dual intersection complex is that it involves a choice of an snc model. The snc models are highly nonunique: we can keep blowing up to get a directed set of snc models. One would like to extract intrinsic information about the degeneration family. There are two general strategies. First, one can analyze the relation between different models and seek an optimal choice using the minimal model program [36][37]; this usually leaves the snc world, and even the optimal choices may still be nonunique. Alternatively we can consider all (snc) models simultaneously, by the language of NA geometry. Good references can be found in [29, A] [3, Appendix][2, 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

|f+g|_{0}\leq\max\{|f|_{0},|g|_{0}\}

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 [29].

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:

•

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 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 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 $X_{K}^{an}$ .
•

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 monomial valuation with the same value for $-\log|z_{j}|_{x},j\in J$ .

009S

Remark 3.1. The retraction map depends on the choice of the model. There are examples where two models $\mathcal{X}$ and $\mathcal{X}^{\prime}$ define the same $\Delta_{\mathcal{X}}$ as a subset of $X_{K}^{an}$ , but the retraction maps are different [21, Appendix].

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:

X_{K}^{an}\simeq\varprojlim_{\text{snc models}}\Delta_{\mathcal{X}}

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

While dual intersection complexes depend strongly on the model, in the CY case the embedding image of the essential skeleton $Sk(\mathcal{X})\subset X_{K}^{an}$ as a set is independent of the model, so can be written as $Sk(X)$ . This can be expected as $Sk(\mathcal{X})$ should support the limiting normalised CY measure, a property independent of the model choice. However, as we blow up snc models, the essential skeleton as a simplical complex can be subdivided.

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||}.

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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.