Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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72 SHINICHI MOCHIZUKI Definition 3.6. Fix a collection of initial Θ-data (F/F, X F ,l,C K , V, V bad mod ,ɛ) as in Definition 3.1. In the following, we shall use the various notations introduced in Definition 3.1 for various objects associated to this initial Θ-data. Then we define aΘ-Hodge theater [relative to the given initial Θ-data] to be a collection of data that satisfies the following conditions: † HT Θ =({ † F v } v∈V , † F ⊩ mod) (a) If v ∈ V non ,then † F v is a category which admits an equivalence of categories † ∼ F →Fv v [where F v is as in Examples 3.2, (i); 3.3, (i)]. In particular, † F v admits a natural Frobenioid structure [cf. [FrdI], Corollary 4.11, (iv)], which may be constructed solely from the category-theoretic structure of † F v .Write † D v , † Dv ⊢ , † Dv Θ , † Fv ⊢ , † Fv Θ for the objects constructed category-theoretically from † F v that correspond to the objects without a “†” discussed in Examples 3.2, 3.3 [cf., especially, Examples 3.2, (vi); 3.3, (iii)]. (b) If v ∈ V arc ,then † F v is a collection of data ( † C v , † D v , † κ v )—where † C v is a category equivalent to the category C v of Example 3.4, (i); † D v is an Aut-holomorphic orbispace; and † κ v : O ⊲ ( † C v ) ↩→ A† D v is an inclusion of topological monoids, which we shall refer to as the Kummer structure on † C v — such that there exists an isomorphism of collections of data † ∼ F →Fv v [where F v is as in Example 3.4, (i)]. Write † Dv ⊢ , † Dv Θ , † Fv ⊢ , † Fv Θ for the objects constructed algorithmically from † F v that correspond to the objects without a “†” discussed in Example 3.4, (ii), (iii). (c) † F ⊩ mod is a collection of data ( † C ⊩ mod, Prime( † C ⊩ mod) ∼ → V, { † F ⊢ v } v∈V , { † ρ v } v∈V ) —where † Cmod ⊩ is a category which admits an equivalence of categories † Cmod ⊩ ∼ →Cmod ⊩ [which implies that † Cmod ⊩ admits a natural category-theoretically constructible Frobenioid structure — cf. [FrdI], Corollary 4.11, (iv); [FrdI], Theorem 6.4, (i)]; Prime( † Cmod ⊩ ) → ∼ V is a bijection of sets, where we write Prime( † Cmod ⊩ ) for the set of primes constructed from the category † Cmod ⊩ [cf. [FrdI], Theorem 6.4, (iii)]; † Fv ⊢ is as discussed in (a), (b) above; † ∼ ρ v :Φ† C ⊩ → Φ mod ,v rlf † C [where we use notation as in the v ⊢ discussion of Example 3.5, (i)] is an isomorphism of topological monoids. Moreover, we require that there exist an isomorphism of collections of data † F ⊩ ∼ mod → F ⊩ mod [where F⊩ mod is as in Example 3.5, (ii)]. Write † F ⊩ tht , † F ⊩ D for the objects constructed algorithmically from † F ⊩ mod that correspond to the objects without a “†” discussed in Example 3.5, (ii), (iii). Remark 3.6.1. When we discuss various collections of Θ-Hodge theaters, labeled by some symbol “□” inplaceofa“†”, we shall apply the notation of Definition 3.6
INTER-UNIVERSAL TEICHMÜLLER THEORY I 73 with “†” replaced by “□” to denote the various objects associated to the Θ-Hodge theater labeled by “□”. Remark 3.6.2. If † HT Θ and ‡ HT Θ are Θ-Hodge theaters, then there is an evident notion of isomorphism of Θ-Hodge theaters † HT Θ ∼ → ‡ HT Θ [cf. Remark 3.5.2]. We leave the routine details to the interested reader. Corollary 3.7. (Θ-Links Between Θ-Hodge Theaters) Fix a collection of initial Θ-data (F/F, X F , l, C K , V, V bad mod , ɛ) as in Definition 3.1. Let † HT Θ =({ † F v } v∈V , † F ⊩ mod); ‡ HT Θ =({ ‡ F v } v∈V , ‡ F ⊩ mod) be Θ-Hodge theaters [relative to the given initial Θ-data]. Then: (i) (Θ-Link) The full poly-isomorphism [cf. §0] between collections of data [cf. Remark 3.5.2] † F ⊩ ∼ tht → ‡ F ⊩ mod is nonempty [cf. Remark 3.7.1 below]. We shall refer to this full poly-isomorphism as the Θ-link † HT Θ Θ −→ ‡ HT Θ from † HT Θ to ‡ HT Θ . (ii) (Preservation of “D ⊢ ”) Let v ∈ V. Recall the tautological isomorphisms □ Dv ⊢ ∼ → □ Dv Θ for □ = †, ‡ — i.e., which arise from the definitions when v ∈ V good , and which arise from a natural product functor [cf. Example 3.2, (v)] when v ∈ V bad . Then we obtain isomorphisms † D ⊢ v ∼ → † D Θ v ∼ → ‡ D ⊢ v by composing the tautological isomorphism just mentioned with any isomorphism induced by a Θ-link isomorphism as in (i). (iii) (Preservation of “O × ”) Let v ∈ V. Recall the tautological isomorphisms O ×□ ∼ →O ×□ [where we omit the notation “(−)”] for □ = †, ‡ — i.e., Cv ⊢ Cv Θ which arise from the definitions when v ∈ V good [cf. Examples 3.3, (ii); 3.4, (iii)], and which are induced by the natural product functor [cf. Example 3.2, (v)] when v ∈ V bad . Then, relative to the corresponding composite isomorphism of (ii), we obtain a composite isomorphism O × † C ⊢ v ∼ → O × † C Θ v ∼ → O × ‡ C ⊢ v by composing the tautological isomorphism just mentioned with any isomorphism induced by a Θ-link isomorphism as in (i). Proof. The various assertions of Corollary 3.7 follow immediately from the definitions and the discussion of Examples 3.2, 3.3, 3.4, and 3.5. ○
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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