Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 107
[where n ∈ Z] isaninfinite chain of D-NF-linked D-ΘNF-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.
(iii) If we regard each of the D-ΘNF-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-ΘNF-Hodge-theaters —asinFig. 4.7
below [cf. the situation discussed in Corollary 3.9, (i)]. In Fig. 4.7, “> ⊢ ” denotes
the mono-analytic core; “/ ↩→ / / ↩→ ...” denotes the “holomorphic” processions
of Proposition 4.11, (i), together with the remaining [“holomorphic”] data of
the corresponding D-ΘNF-Hodge theater. 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-ΘNF-Hodge-theaters].
/ ↩→ / / ↩→ ...
...
|
...
/ ↩→ / / ↩→ ...
—
> ⊢
—
/ ↩→ / / ↩→ ...
...
|
...
/ ↩→ / / ↩→ ...
Fig. 4.7: Étale-picture of D-ΘNF-Hodge theaters