Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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130 SHINICHI MOCHIZUKI (vi) Let † D ⊚± be any category isomorphic to D ⊚± . Then just as in the discussion of (iii) in the case of “v ∈ V good ⋂ V non ”, it makes sense [cf. [AbsTopI], Lemma 4.5] to speak of the set of cusps of † D ⊚± ,aswellastheset of ±-label classes of cusps LabCusp ± ( † D ⊚± ) — which, in this case, may be identified with the set of cusps of † D ⊚± . (vii) Recall from [AbsTopIII], Theorem 1.9 [cf. Remark 3.1.2] that [just as in the case of D ⊚ — cf. the discussion of Definition 4.1, (v)] there exists a grouptheoretic algorithm for reconstructing, from π 1 (D ⊚± )[cf.§0], the algebraic closure “F ” of the base field “K”, hence also the set of valuations “V(F )” from D ⊚± [e.g., as a collection of topologies on F — cf., e.g., [AbsTopIII], Corollary 2.8]. Moreover, for w ∈ V(K) arc , let us recall [cf. Remark 3.1.2; [AbsTopIII], Corollaries 2.8, 2.9] that one may reconstruct group-theoretically, fromπ 1 (D ⊚± ), the Aut-holomorphic orbispace X w associated to X w .Let † D ⊚± be as in (vi). Then let us write V( † D ⊚± ) for the set of valuations [i.e., “V(F )”], equipped with its natural π 1 ( † D ⊚± )-action, V( † D ⊚± ) def = V( † D ⊚± )/π 1 ( † D ⊚± ) for the quotient of V( † D ⊚± )byπ 1 ( † D ⊚± ) [i.e., “V(K)”], and, for w ∈ V( † D ⊚± ) arc , X( † D ⊚± ,w) [i.e., “X w ” — cf. the discussion of [AbsTopIII], Definition 5.1, (ii)] for the Autholomorphic orbispace obtained by applying these group-theoretic reconstruction algorithms to π 1 ( † D ⊚± ). Now if U is an arbitrary Aut-holomorphic orbispace, then let us define a morphism U → † D ⊚± to be a morphism of Aut-holomorphic orbispaces [cf. [AbsTopIII], Definition 2.1, (ii)] U → X( † D ⊚± ,w)forsomew ∈ V( † D ⊚± ) arc . Thus, it makes sense to speak of the pre-composite (respectively, post-composite) of such a morphism U → † D ⊚± with a morphism of Aut-holomorphic orbispaces (respectively, with an isomorphism [cf. §0] † D ⊚± ∼ → ‡ D ⊚± [i.e., where ‡ D ⊚± is a category equivalent to D ⊚± ]). Remark 6.1.1. In fact, in the notation of Example 4.3, (i); Definition 6.1, (v), it is not difficult to verify that V ± = V ±un (⊆ V(K)). Example 6.2. Model Base-Θ ± -Bridges. (i) In the following, let us think of F l as a F ± l -group [relative to the evident F ± l -group structure]. Let D ≻ = {D ≻,v } v∈V ; D t = {D vt } v∈V
INTER-UNIVERSAL TEICHMÜLLER THEORY I 131 —wheret ∈ F l , and we use the notation v t to denote the pair (t, v) [cf. Example 4.3, (iv)] — be copies of the “tautological D-prime-strip” {D v } v∈V [cf. Examples 4.3, (iv); 4.4, (ii)]. For each t ∈ F l ,write φ Θ± v t : D vt →D ≻,v ; φ Θ± t : D t → D ≻ for the respective positive +-full poly-isomorphisms, i.e., relative to the respective identifications with the “tautological D-prime-strip” {D v } v∈V . Write D ± for the capsule {D t } t∈Fl [cf. the constructions of Example 4.4, (iv)] and φ Θ± ± : D ± → D ≻ for the collection of poly-morphisms {φ Θ± t } t∈Fl . (ii) The collection of data (D ± , D ≻ ,φ Θ± ± ) admits a natural poly-automorphism of order two −1 Fl defined as follows: the poly-automorphism −1 Fl acts on F l as multiplication by −1 and induces the polyisomorphisms D t → D−t [for t ∈ F l ]andD ≻ → D≻ determined [i.e., relative to ∼ ∼ the respective identifications with the “tautological D-prime-strip” {D v } v∈V ]by the +-full poly-automorphism whose sign at every v ∈ V is negative. One verifies immediately that −1 Fl , defined in this way, is compatible [in the evident sense] with φ Θ± ± . (iii) Let α ∈{±1} V .Thenα determines a natural poly-automorphism of order two α Θ± of the collection of data (D ± , D ≻ ,φ Θ± ± ) as follows: the poly-automorphism α Θ± acts on F l as the identity and on D t ,for t ∈ F l ,andD ≻ as the α-signed +-full poly-automorphism. One verifies immediately that α Θ± , defined in this way, is compatible [in the evident sense] with φ Θ± ± . Example 6.3. Model Base-Θ ell -Bridges. F ± l (i) In the following, let us think of F l as a F ± l -torsor structure]. Let D t = {D vt } v∈V -torsor [relative to the evident [for t ∈ F l ]andD ± be as in Example 6.2, (i); D ⊚± as in Definition 6.1, (v). In the following, let us fix an isomorphism of F ± l -torsors LabCusp ± (D ⊚± ) ∼ → F l [cf. the discussion of Definition 6.1, (v)], which we shall use to identify LabCusp ± (D ⊚± ) with F l . Note that this identification induces an isomorphism of groups Aut ± (D ⊚± )/Aut csp (D ⊚± ) ∼ → F ⋊± l
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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