Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 89
§I5]. In fact, [although this point of view is not mentioned in [AbsTopIII]] one may
“compose” this analogy with the analogy between the p-adic and complex theories
discussed in [pOrd], Introduction; [pTeich], Introduction, §0, and consider the
analogy between the mono-anabelian theory of [AbsTopIII] and the classical geometry
of the upper half-plane H. In addition to being more elementary than
the p-adic theory, this analogy with the classical geometry of the upper half-plane
H also has the virtue that
since it revolves around the canonical Kähler metric — i.e., the Poincaré
metric— on the upper half-plane, it renders more transparent the
relationship between the theory of the present series of papers and classical
Arakelov theory [which also revolves, to a substantial extent, around
Kähler metrics at the archimedean primes].
(ii) The essential content of the mono-anabelian theory of [AbsTopIII] may be
summarized by the diagram
Π k ×
log
−→ k Π (∗)
—wherek is a finite extension of Q p ; k is an algebraic closure of k; Π is the arithmetic
fundamental group of a hyperbolic orbicurve over k; log is the p-adic logarithm
[cf. [AbsTopIII], §I1]. On the other hand, if (E, ∇ E ) denotes the “tautological
indigenous bundle” on H [i.e., the first de Rham cohomology of the tautological
elliptic curve over H], then one has a natural Hodge filtration 0 → ω →E→τ → 0
[where ω, τ def
= ω −1 are holomorphic line bundles on H], together with a natural
complex conjugation operation ι E : E→E. The composite
ω ↩→ E
ι E
−→ E ↠ τ
then determines an Hermitian metric |−| ω on ω. For any trivializing section f of
ω, the(1, 1)-form
def
κ H = 1
2πi ∂∂ log(|f| ω)
is the canonical Kähler metric [i.e., Poincaré metric] on H. Then one can already
readily identify various formal similarities between κ H and the diagram (∗)
reviewed above: Indeed, at a purely formal [but by no means coincidental!] level,
the “log” that appears in the definition of κ H is reminiscent of the “log-Frobenius
operation” log. At a less formal level, the “Galois group” Π is reminiscent — cf. the
point of view that “Galois groups are arithmetic tangent bundles”, a point of view
that underlies the theory of the arithmetic Kodaira-Spencer morphism discussed in
[HASurI]! — of ∂. If one thinks of complex conjugation as a sort of “archimedean
Frobenius” [cf. [pTeich], Introduction, §0], then ∂ is reminiscent of the “Galois
group” Π operating on the opposite side [cf. ι E ] of the log-Frobenius operation
log. The Hodge filtration of E corresponds to the ring structures of the copies of
k on either side of log [cf. the discussion of [AbsTopIII], Remark 3.7.2]. Finally,
perhaps most importantly from the point of view of the theory of the present series
of papers: