1.6 Affinoid torus fibrations and integral affine structures [04MW]

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1.6 Affinoid torus fibrations and integral affine structures

Let $X$ be a smooth proper variety over $K$ .

Definition 1.6.1.

Let $\rho:X^{\text{an}}\longrightarrow B$ be a continuous map to a topological space $B$ . For any point $b\in B$ , we say that $\rho$ is an affinoid torus fibration at $b$ if there exists an open neighbourhood $U$ of $b$ in $B$ , such that the restriction to $\rho^{-1}(U)$ fits into a commutative diagram:

$V$ being an open subset of $\mathbb{R}^{n}$ , the upper horizontal map an isomorphism of analytic spaces, the lower horizontal map a homeomorphism, and the map $\val$ defined as in Section 1.5.

Example 1.6.2.

It follows from the definition of good dlt model $\mathscr{X}$ of $X$ that the Berkovich retraction $\rho_{\mathscr{X}}:X^{\an}\rightarrow\Sk(\mathscr{X})$ is an affinoid torus fibration over the interior of the maximal faces $\tau$ of $\Sk(\mathscr{X})$ . Indeed, the retraction over $\textrm{Int}({\tau})$ only depends on the formal completion of $\mathscr{X}$ along the corresponding 0-dimensional stratum $p$ . The pair $(\mathscr{X},\mathscr{X}_{k})$ is snc at $p$ , hence the claim.

Example 1.6.3.

If $\mathscr{X}/R$ is a toric model of $X=\mathbb{T}$ , it follows from Proposition 1.5.2 that the Berkovich retraction:

\rho_{\mathscr{X}}:\widehat{\mathscr{X}}_{\eta}\longrightarrow\Sk(\mathscr{X})

is an affinoid torus fibration over the interior of $\Sk(\mathscr{X})$ . This also holds when $X$ is a regular proper toric variety over $K$ , and $\mathscr{X}$ a regular proper toric model, by [GJKM19, Theorem A.4].

Note that the above definition implies that $B$ is a topological manifold at $b$ ; in the case of a Berkovich retraction $\rho_{\mathscr{X}}$ , this does not necessarily hold at every point of $\Sk(\mathscr{X})$ .

Given a continuous map $\rho:X^{\text{an}}\longrightarrow B$ , we denote by $B^{\textrm{sm}}$ the locus of points in $B$ where $\rho$ is an affinoid torus fibration at; we call $B\setminus B^{\textrm{sm}}$ the discriminant or singular locus of $B$ . $B^{\textrm{sm}}$ is endowed with an integral affine structure; we recall the definition and describe such structure.

Definition 1.6.4.

An integral affine structure on a topological manifold is an atlas of charts with transition functions in $\textrm{GL}_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$ .

Definition 1.6.5.

An integral affine function on an open subset of $\mathbb{R}^{n}$ is a continuous real-valued function locally of the form $f(x_{1},\ldots,x_{n})=a_{1}x_{1}+\ldots+a_{n}x_{n}+b$ , with $a_{i}\in\mathbb{Z}$ and $b\in\mathbb{R}$ . We denote by $\textrm{Aff}_{\mathbb{R}^{n}}$ the sheaf of integral affine functions on $\mathbb{R}^{n}$ .

Lemma 1.6.6 ([KS06, 2.1]).

An integral affine structure on a topological manifold $M$ is equivalent to the datum of a subsheaf $\mathrm{Aff}_{M}$ of the sheaf of continuous functions on $M$ such that $(M,\mathrm{Aff}_{M})$ is locally isomorphic to $(\mathbb{R}^{n},\mathrm{Aff}_{\mathbb{R}^{n}})$ .

If $\rho$ is an affinoid torus fibration over $B^{\textrm{sm}}\subseteq B$ , the integral affine structure on $B^{\textrm{sm}}$ is the pull-back of $\textrm{Aff}_{\mathbb{R}^{n}}$ via the charts in Definition 1.6.1. An alternative description of this structure is given in [KS06, 4.1, Theorem 1]: let $U\subset B^{\textrm{sm}}$ be a connected open subset. Then if $h$ is an invertible analytic function on $\rho^{-1}(U)$ , its modulus $\lvert h\rvert$ is constant on the fibers of $\rho$ by the maximum principle, so that it defines a continuous function on the base. We now have:

\mathrm{Aff}_{B^{\textrm{sm}}}(U)=\{-\log\lvert h\rvert\,|\,h\in\mathcal{O}^{\times}_{X^{\an}}(\rho^{-1}(U))\}.

Remark 1.6.7.

Given an integral affine structure on a topological manifold $M$ , there is a monodromy representation

T:\pi_{1}(M)\rightarrow\textrm{GL}_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}

defined by covering a loop in $M$ by affine charts and composing the corresponding transition functions. See [KS06, 2.2] for more details.