Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

76 SHINICHI MOCHIZUKI
rise to an isomorph of Dv
⊢ that is independent of n”. In particular, the various
isomorphs of Dv ⊢ associated to the copies of −→v X that arise from n HT Θ for different
n are only related to one another via some indeterminate isomorphism. Thus, from
the point of view of the theory of [AbsTopIII] [cf. [AbsTopIII], §I3; [AbsTopIII],
Remark 5.10.2, (ii)], each −→v X gives rise to a well-defined ring structure — i.e., a
“holomorphic structure” —whichisobliterated by the indeterminate isomorphism
between the isomorphs of Dv ⊢ arising from n HT Θ for distinct n.
The discussion of Remark 3.8.1, (iii), (iv), may be summarized as follows.
Corollary 3.9. (Étale-pictures of Θ-Hodge Theaters) In the situation of
Corollary 3.8, let v ∈ V. Then:
n D v
... |
...
|
(n−1) D v
— — D ⊢ v
— — (n+1)
D v
...
|
|
...
(n+2) D v
Fig. 3.2:
Étale-picture of Θ-Hodge theaters
(i) We have a diagram as in Fig. 3.2, which we refer to as the étale-picture.
Here, each horizontal and vertical “— —” denotes the relationship between (−) D v
and Dv ⊢ — i.e., an extension of topological groups when v ∈ V non , or the underlying
object of TM ⊢ arising from the associated topological field when v ∈ V arc —discussed
in Remark 3.8.1, (iii), (iv). [Unlike the Frobenius-picture!] the étale-picture
admits arbitrary permutation symmetries among the labels n ∈ Z corresponding
to the various Θ-Hodge theaters. Put another way, the étale-picture may be
thought of as a sort of canonical splitting of the Frobenius-picture.
(ii) In a similar vein, we have a diagram as in Fig. 3.3 below, obtained
by replacing the “Dv ⊢ ” in the middle of Fig. 3.2 by “Dv ⊢ O × ”. Here, each
Cv ⊢
horizontal and vertical “— —” denotes the relationship between (−) D v and Dv
⊢
discussed in (i); when v ∈ V non , the notation “Dv ⊢ O × ” denotes an isomorph
Cv ⊢
of the pair consisting of the category D ⊢ v
together with the group-like monoid O × C ⊢ v