Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

74 SHINICHI MOCHIZUKI
Remark 3.7.1. One verifies immediately that there exist many distinct isomorphisms
† F ⊩ ∼
tht
→ ‡ F ⊩ mod
as in Corollary 3.7, (i), none of which is conferred a “distinguished”
status, i.e., in the fashion of the “natural isomorphism F ⊩ ∼
mod
→ F ⊩ tht ”
discussed in Example 3.5, (ii).
The following result follows formally from Corollary 3.7.
Corollary 3.8. (Frobenius-pictures of Θ-Hodge Theaters) Fix a collection
of initial Θ-data as in Corollary 3.7. Let { n HT Θ } n∈Z be a collection of distinct
Θ-Hodge theaters indexed by the integers. Then by applying Corollary 3.7, (i),
with † HT Θ def
= n HT Θ , ‡ HT Θ def
= (n+1) HT Θ , we obtain an infinite chain
...
Θ
−→ (n−1) HT Θ Θ
−→ n HT Θ Θ
−→ (n+1) HT Θ Θ
−→ ...
of Θ-linked Θ-Hodge theaters. This infinite chain may be represented symbolically
as an oriented graph ⃗ Γ [cf. [AbsTopIII], §0]
... → • → • → • → ...
— i.e., where the arrows correspond to the “ −→’s”, Θ
and the “•’s” correspond to the
“ n HT Θ ”. This oriented graph ⃗ Γ admits a natural action by Z — i.e., a translation
symmetry — but it does not admit arbitrary permutation symmetries. For
instance, ⃗ Γ does not admit an automorphism that switches two adjacent vertices,
but leaves the remaining vertices fixed. Put another way, from the point of view
of the discussion of [FrdI], Introduction, the mathematical structure constituted by
this infinite chain is “Frobenius-like”, or “order-conscious”. It is for this
reason that we shall refer to this infinite chain in the following discussion as the
Frobenius-picture.
Remark 3.8.1.
(i) Perhaps the central defining aspect of the Frobenius-picture is the fact that
the Θ-link maps
n Θ v
↦→ (n+1) q
v
[i.e., where v ∈ V bad — cf. the discussion of Example 3.2, (v)].
...
----
n q
v
n Θ v
----
(n+1) q
v
(n+1) Θ v
----
...
n Θ v
↦→ (n+1) q
v
Fig. 3.1: Frobenius-picture of Θ-Hodge theaters