Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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132 SHINICHI MOCHIZUKI [cf. the discussion of Definition 6.1, (v)], which we shall use to identify the group Aut ± (D ⊚± )/Aut csp (D ⊚± ) with the group F ⋊± l . If v ∈ V good ⋂ V non (respectively, v ∈ V bad ), then the natural restriction functor on finite étale coverings arising from the natural composite morphism −→v X → X v → X K (respectively, X v → X v → X K ) determines [cf. Examples 3.2, (i); 3.3, (i)] a natural morphism φ Θell •,v : D v → D ⊚± [cf. the discussion of Example 4.3, (ii)]. If v ∈ V arc , then [cf. Example 3.4, (i)] ∼ we have a tautological morphism D v = −→v X → X v → X(D ⊚± ,v), hence a morphism φ Θell •,v : D v →D ⊚± [cf. the discussion of Example 4.3, (iii)]. For arbitrary v ∈ V, write φ Θell v 0 : D v0 →D ⊚± for the poly-morphism given by the collection of morphisms D v0 β ◦ φ Θell •,v ◦ α →D ⊚± of the form —whereα ∈ Aut + (D v0 ); β ∈ Aut csp (D ⊚± ); we apply the tautological identification of D v with D v0 [cf. the discussion of Example 4.3, (ii), (iii), (iv)]. Write φ Θell 0 : D 0 →D ⊚± for the poly-morphism determined by the collection {φ Θell v 0 : D v0 →D ⊚± } v∈V [cf. the discussion of Example 4.3, (iv)]. Note that the existence of “β” in the definition of φ Θell v 0 implies that it makes sense to post-compose φ Θell 0 with an element of Aut ± (D ⊚± )/Aut csp (D ⊚± ) → ∼ F ⋊± l .Thus,foranyt ∈ F l ⊆ F ⋊± l ,letuswrite φ Θell t : D t →D ⊚± for the result of post-composing φ Θell 0 with the “poly-action” [i.e., action via polyautomorphisms] of t on D ⊚± [and pre-composing with the tautological identification of D 0 with D t ]and φ Θell ± : D ± →D ⊚± for the collection of arrows {φ Θell t } t∈Fl . (ii) Let γ ∈ F ⋊± l . Then γ determines a natural poly-automorphism γ ± of D ± as follows: the automorphism γ ± acts on F l via the usual action of F ⋊± l on F l and, ∼ for t ∈ F l , induces the +-full poly-isomorphism D t → Dγ(t) whose sign at every v ∈ V is equal to the sign of γ [cf. the construction of Example 6.2, (ii)]. Thus, we obtain a natural poly-action of F ⋊± l on D ± . On the other hand, the isomorphism Aut ± (D ⊚± )/Aut csp (D ⊚± ) → ∼ F ⋊± l of (i) determines a natural poly-action of F ⋊± l on D ⊚± . Moreover, one verifies immediately that φ Θell ± is equivariant with respect to these poly-actions of F ⋊± l on D ± and D ⊚± ; in particular, we obtain a natural poly-action F ⋊± l (D ± , D ⊚± ,φ Θell ± ) of F ⋊± l on the collection of data (D ± , D ⊚± ,φ Θell ± ) [cf. the discussion of Example 4.3, (iv)].
INTER-UNIVERSAL TEICHMÜLLER THEORY I 133 Definition 6.4. In the following, we shall write l ± def = l +1=(l +1)/2. (i) We define a base-Θ ± -bridge, orD-Θ ± -bridge, [relative to the given initial Θ-data] to be a poly-morphism † † φ Θ± ± D T −→ † D ≻ —where † D ≻ is a D-prime-strip; T is an F ± l -group; † D T = { † D t } t∈T is a capsule of D-prime-strips, indexed by [the underlying set of] T — such that there exist isomorphisms D ≻ ∼ → † D ≻ , D ± ∼ → † D T ∼ — where we require that the bijection of index sets F l → T induced by the second isomorphism determine an isomorphism of F ± l -groups — conjugation by which maps φ Θ± ± ↦→ † φ Θ± ± . In this situation, we shall write † D |T | for the l ± -capsule obtained from the l-capsule † D T by forming the quotient |T | of the index set T of this underlying capsule by the action of {±1} and identifying the components of the capsule † D T indexed by the elements in the fibers of the quotient T ↠ |T | via the constituent poly-morphisms of † φ Θ± ± = { † φ Θ± t } t∈Fl [so each consitutent D-prime-strip of † D |T | is only well-defined up to a positive automorphism, but this indeterminacy will not affect applications of this construction — cf. Propositions 6.7; 6.8, (ii); 6.9, (i), below]. Also, we shall write † D T for the l -capsule determined by the subset T def = |T |\{0} of nonzero elements of |T |. We define a(n) [iso]morphism of D-Θ ± -bridges to be a pair of poly-morphisms ( † † φ Θ± ± D T −→ † D ≻ ) → ( ‡ D T ′ ‡ φ Θ± ± −→ ‡ D ≻ ) † D T ∼ → ‡ D T ′; † D ≻ ∼ → ‡ D ≻ —where † ∼ D T → ‡ D T ′ is a capsule-+-full poly-isomorphism whose induced morphism on index sets T → ∼ T ′ is an isomorphism of F ± l -groups; † ∼ D ≻ → ‡ D ≻ is a +-full poly-isomorphism — which are compatible with † φ Θ± ± , ‡ φ Θ± ± . There is an evident notion of composition of morphisms of D-Θ ± -bridges. (ii) We define a base-Θ ell -bridge [i.e., a “base-Θ-elliptic-bridge”], or D-Θ ell - bridge, [relative to the given initial Θ-data] to be a poly-morphism † † φ Θell ± D T −→ † D ⊚± —where † D ⊚± is a category equivalent to D ⊚± ; T is an F ± l -torsor; † D T = { † D t } t∈T is a capsule of D-prime-strips, indexed by [the underlying set of] T — such that there exist isomorphisms D ⊚± ∼ → † D ⊚± , D ± ∼ → † D T
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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