Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

90 SHINICHI MOCHIZUKI
the fact that log-shells play the role in the theory of [AbsTopIII] of “canonical
rigid integral structures” [cf. [AbsTopIII], §I1] — i.e., “canonical standard
units of volume” — is reminiscent of the fact that the Kähler metric
κ H also plays the role of determining a canonical notion of volume on H.
(iii) From the point of view of the analogy discussed in (ii), property (a) of
Remark 4.3.2 may be thought of as corresponding to the local representability
via the [positive] (1, 1)-form κ H —on,say,acompactquotientS of H —of
the [positive] global degree of [the result of descending to S] the line bundle ω;
property (b) of Remark 4.3.2 may be thought of as corresponding to the fact that
this (1, 1)-form κ H that gives rise to a local representation on S of the notion of
a positive global degree not only exists locally on S, but also admits a canonical
global extension to the entire Riemann surface S which may be related to the
algebraic theory [i.e., of algebraic rational functions on S].
(iv) The analogy discussed in (ii) may be summarized as follows:
mono-anabelian theory
geometry of the upper-half plane H
the Galois group Π
the differential operator ∂
the Galois group Π
the differential operator
on the opposite side of log
∂
the ring structures of the copies the Hodge filtration of E,
of k on either side of log
ι E , |−| E
log-shells as
the canonical Kähler volume
canonical units of volume
κ H
Example 4.4. Model Base-Θ-Bridges. In the following, we construct the
“models” for the notion of a “base-Θ-bridge” [cf. Definition 4.6, (ii), below]. We
continue to use the notation of Example 4.3.
(i) Let v ∈ V bad . Recall that there is a natural bijection between the set
|F l |
def
= F l /{±1} =0 ⋃ F l
[i.e., the set of {±1}-orbits of F l ] and the set of cusps of the hyperbolic orbicurve
C v [cf. [EtTh], Corollary 2.9]. Thus, [by considering fibers over C v ] we obtain
labels ∈|F l | of various collections of cusps of X v , X v
.Write
μ − ∈ X v (K v )
for the unique torsion point of order 2 whose closure in any stable model of X v
over O Kv intersects the same irreducible component of the special fiber of the stable
model as the [unique] cusp labeled 0 ∈|F l |. Now observe that it makes sense to
speak of the points ∈ X v (K v ) obtained as μ − -translates of the cusps, relativetothe