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A.4. Hybrid geometry over ℂ [018I]

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A.4. Hybrid geometry over ℂ{\mathbb{C}}

If XX is a scheme of finite type over ℂ{\mathbb{C}}, we denote by Xhol=X⁡(ℂ)X^{\operatorname{hol}}=X({\mathbb{C}}) its analytification with respect to the usual absolute value |⋅|∞|\cdot|_{\infty}, by X0anX^{\mathrm{an}}_{0} its analytification with respect to the trivial absolute value, and by XhybX^{\mathrm{hyb}} its analytification with respect to the hybrid norm ∥⋅∥hyb\|\cdot\|_{\mathrm{hyb}}.

From the structure morphism X→Spec⁡ℂX\to\operatorname{Spec}{\mathbb{C}} we obtain a continuous map λ:Xhyb→ℳ⁡(ℂhyb)≃[0,1]\lambda\colon X^{\mathrm{hyb}}\to{\mathcal{M}}({\mathbb{C}}_{\mathrm{hyb}})\simeq[0,1]. The fiber λ−1​(ρ)\lambda^{-1}(\rho) is equal to the analytification of XX with respect to the multiplicative norm |⋅|∞ρ|\cdot|_{\infty}^{\rho} on ℂ{\mathbb{C}}. In particular, we have canonical identifications λ−1​(1)≃Xhol\lambda^{-1}(1)\simeq X^{\operatorname{hol}} and λ−1​(0)≃X0An\lambda^{-1}(0)\simeq X^{\mathrm{An}}_{0}. For 0<ρ≤10<\rho\leq 1, the fiber λ−1​(ρ)\lambda^{-1}(\rho) is also homeomorphic to XholX^{\operatorname{hol}}. In fact, we have a a homeomorphism

λ−1​((0,1])≃(0,1]×Xhol,\lambda^{-1}\left((0,1]\right)\simeq(0,1]\times X^{\operatorname{hol}},

see [Berk09, Lemma 2.1].

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