2.6 Integral affine structure and toric irreducible components [04P6]

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2.6 Integral affine structure and toric irreducible components

The case where $Z$ is an irreducible component $D$ of $\mathscr{X}_{k}$ is particularly relevant for proving Theorem A. Under the assumptions of Theorem B, we proved that $\mathscr{X}$ is toric along $D$ , and $\rho_{\mathscr{X}}$ is an affinoid torus fibration over $\Star(v_{D})$ by Corollary C. Moreover, we have the following explicit description of the $\mathbb{Z}$ -affine structure on $\Star(v_{D})$ induced by $\rho_{\mathscr{X}}$ - note that it only depends on $D$ and not on how $D$ sits inside $\mathscr{X}$ .

Corollary 2.6.1.

In the setting of Theorem B, let $Z=D$ be an irreducible component of $\mathscr{X}_{k}$ . Then there is a natural $\mathbb{Z}$ -linear embedding of $\Star(v_{D})$ inside the fan $\Sigma_{D}$ of $D$ which sends the polyhedral decomposition of $\Star(v_{D})$ to the cone decomposition of $\Sigma_{D}$ .

Proof.

By the proof of Theorem B and Proposition 1.5.2 we have the following diagram:

where the upper arrow is an isomorphism of analytic spaces, and the lower one a homeomorphism. Here $\mathfrak{X}_{D}$ and $\mathfrak{N}_{D}$ are the generic fibers (in the sense of Berkovich) of the formal completions $\widehat{\mathscr{X}_{/D}}$ and $\widehat{\mathscr{N}_{/D}}$ respectively, and $\mathrm{Int}(\Sigma_{1})$ denotes the interior of the polyhedral complex $\Sigma_{1}$ obtained by intersecting the fan $\hat{\Sigma}\subset N_{\mathbb{R}}\times\mathbb{R}$ of the normal bundle of $D$ in $\mathscr{X}$ with $N_{\mathbb{R}}\times\{1\}$ . In particular, $\mathrm{Int}(\Sigma_{1})$ is embedded in $\Sigma_{D}\simeq N_{\mathbb{R}}\simeq\mathbb{R}^{n}$ , the polyhedral decomposition of $\Star(v_{D})$ is the same of $\Sigma_{D}$ , and the vertex $v_{D}$ corresponds to the origin. By Section 1.6, the integral affine structure on $\Star(v_{D})$ is the pullback via $\varphi$ of the integral affine structure on $\Sigma_{D}$ , and this concludes the proof. ∎