Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

128 SHINICHI MOCHIZUKI
—whichiswell-defined up to multiplication by ±1 andcompatible, relativetothe
natural bijection to “LabCusp(−)” of the preceding display, with the natural bijection
of the second display of Proposition 4.2. That is to say, in the terminology of
(i), LabCusp ± ( † D v ) is equipped with a natural F ± l -group structure. ThisF± l -group
structure determines a natural surjection
Aut( † D v ) ↠ {±1}
— i.e., by considering the induced automorphism of LabCusp ± ( † D v ). Write
Aut + ( † D v ) ⊆ Aut( † D v )
for the index two subgroup of “positive automorphisms” [i.e., the kernel of the above
surjection] and Aut − ( † D v ) def
= Aut( † D v ) \ Aut + ( † D v ) [i.e., where “\” denotes the
set-theoretic complement] for the subset of “negative automorphisms”. In a similar
vein, we shall write
Aut + ( † D) ⊆ Aut( † D)
for the subgroup of “positive automorphisms” [i.e., automorphisms each of whose
components, for v ∈ V, ispositive], and, if α ∈{±1} V [i.e., where we write {±1} V
for the set of set-theoretic maps from V to {±1}],
Aut α ( † D) ⊆ Aut( † D)
for the subset of “α-signed automorphisms” [i.e., automorphisms each of whose
components, for v ∈ V, ispositive if α(v) = +1 and negative if α(v) =−1].
(iv) Suppose that we are in the situation of (ii). Let
‡ D = { ‡ D v } v∈V
be another D-prime-strip [relative to the given initial Θ-data]. Then for any v ∈ V,
we shall refer to as a +-full poly-isomorphism † ∼
D v → ‡ D v any poly-isomorphism
obtained as the Aut + ( † D v )- [or, equivalently, Aut + ( ‡ D v )-] orbit of an isomorphism
† ∼
D v → ‡ D v . In particular, if † D = ‡ D, then there are precisely two +-full polyisomorphisms
† ∼
D v → ‡ D v , namely, the +-full poly-isomorphism determined by the
identity isomorphism, which we shall refer to as positive, and the unique nonpositive
+-full poly-isomorphism, which we shall refer to as negative. In a similar
vein, we shall refer to as a +-full poly-isomorphism † D → ∼ ‡ D any poly-isomorphism
obtained as the Aut + ( † D)- [or, equivalently, Aut + ( ‡ D)-] orbit of an isomorphism
† D → ∼ ‡ D. In particular, if † D = ‡ D, then the set of +-full poly-isomorphisms
† D → ∼ ‡ D is in natural bijective correspondence [cf. the discussion of (iii) above]
with the set {±1} V ; we shall refer to the +-full poly-isomorphism † D → ∼ ‡ D that
corresponds to α ∈ {±1} V as the α-signed +-full poly-isomorphism. Finally, a
capsule-+-full poly-morphism between capsules of D-prime-strips
{ † D t } t∈T
∼
→{ ‡ D t ′} t′ ∈T ′
is defined to be a poly-morphism between two capsules of D-prime-strips determined
by +-full poly-isomorphisms † ∼
D t → ‡ D ι(t) [where t ∈ T ] between the constituent
objects indexed by corresponding indices, relative to some injection ι : T↩→ T ′ .