Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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142 SHINICHI MOCHIZUKI (ii) If ... D −→ (n−1) HT D-Θ±ell D −→ n HT D-Θ±ell D −→ (n+1) HT D-Θ±ell D −→ ... [where n ∈ Z] isaninfinite chain of D-Θ ±ell -linked D-Θ ±ell -Hodge theaters [cf. the situation discussed in Corollary 3.8], then we obtain a resulting chain of full poly-isomorphisms ... ∼ → n D ⊢ > ∼ → (n+1) D ⊢ > ∼ → ... [cf. the situation discussed in Remark 3.8.1, (ii)] between the D ⊢ -prime-strips obtained by applying the functor of (i). That is to say, the output data of the functor of (i) forms a constant invariant [cf. the discussion of Remark 3.8.1, (ii)] — i.e., a mono-analytic core [cf. the situation discussed in Remark 3.9.1] — of the above infinite chain. / ± ↩→ / ± / ± ↩→ ... ... | ... / ± ↩→ / ± / ± ↩→ ... — > ⊢ = {0, ≻} ⊢ — / ± ↩→ / ± / ± ↩→ ... ... | ... / ± ↩→ / ± / ± ↩→ ... Fig. 6.3: Étale-picture of D-Θ±ell -Hodge theaters (iii) If we regard each of the D-Θ ±ell -Hodge theaters of the chain of (ii) as a spoke emanating from the mono-analytic core discussed in (ii), then we obtain a diagram — i.e., an étale-picture of D-Θ ±ell -Hodge-theaters —asin Fig. 6.3 [cf. the situation discussed in Corollary 3.9, (i)]. In Fig. 6.3, “> ⊢ ” denotes the mono-analytic core, obtained [cf. (i); Proposition 6.7] by identifying the mono-analyticized D-prime-strips of the D-Θ ±ell -Hodge theater labeled “0” and “≻”; “/ ± ↩→ / ± / ± ↩→ ...” denotes the “holomorphic” processions of Proposition 6.9, (i), together with the remaining [“holomorphic”] data of the corresponding D- Θ ±ell -Hodge theater. In particular, the mono-analyticizations of the zero-labeled D-prime-strips — i.e., the D-prime-strips corresponding to the first “/ ± ” in the processions just discussed — in the various spokes are identified with one another. Put another way, the coric D ⊢ -prime-strip “> ⊢ ” may be thought of as being equipped with various distinct “holomorphic structures” — i.e., D-prime-strip structures that give rise to the D ⊢ -prime-strip structure — corresponding to the various
INTER-UNIVERSAL TEICHMÜLLER THEORY I 143 spokes. Finally, [cf. the situation discussed in Corollary 3.9, (i)] this diagram satisfies the important property of admitting arbitrary permutation symmetries among the spokes [i.e., among the labels n ∈ Z of the D-Θ ±ell -Hodge-theaters]. (iv) The constructions of (i), (ii), (iii) are compatible, respectively, with the constructions of Corollary 4.12, (i), (ii), (iii), relative to the functor [i.e., determined by the functorial algorithm] of Proposition 6.7, in the evident sense [cf. the compatibility discussed in Proposition 6.9, (iii)]. Finally, we conclude with additive analogues of Definition 5.5, Corollary 5.6. Definition 6.11. (i) We define a Θ ± -bridge [relative to the given initial Θ-data] to be a polymorphism † † ψ Θ± ± F T −→ —where † F ≻ is a F-prime-strip; T is an F ± l -group; † F T = { † F t } t∈T is a capsule of F-prime-strips, indexed by [the underlying set of] T — that lifts a D-Θ ± -bridge † φ Θ± ± : † D T → † D ≻ [cf. Corollary 5.3, (ii)]. In this situation, we shall write † F ≻ † F |T | for the l ± -capsule obtained from the l-capsule † F 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 † F T indexed by the elements in the fibers of the quotient T ↠ |T | via the constituent poly-morphisms of † ψ± Θ± = { † ψt Θ± } t∈Fl [so each consitutent F-prime-strip of † F |T | is only well-defined up to a positive automorphism, but this indeterminacy will not affect applications of this construction — cf. the discussion of Definition 6.4, (i)]. Also, we shall write † F 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 F-Θ ± -bridges ( † † ψ Θ± ± F T −→ † F ≻ ) → ( ‡ F T ′ ‡ ψ Θ± ± −→ ‡ F ≻ ) to be a pair of poly-isomorphisms † F T ∼ → ‡ F T ′; † F ≻ ∼ → ‡ F ≻ that lifts a morphism between the associated D-Θ ± -bridges † φ Θ± ± , ‡ φ Θ± ± .Thereis an evident notion of composition of morphisms of F-Θ ± -bridges. (ii) We define a Θ ell -bridge [relative to the given initial Θ-data] † F T † ψ Θell ± −→ † D ⊚±
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associated to † F ⊩ mod and ‡
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completely determined by † D. INT
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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