On the universal degenerating family over the

リーマン面の退化現象２
（これからのこと）
－普遍退化族？の構成－
松本幸夫（学習院大学理学部）
札幌幾何学セミナー
2009/02/17
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Notation
: a compact Riemann surface of genus
: mapping class group of
: Teichmüller space of
complex analytic space
acts
-analytically
: moduli space
= moduli space of stable curves
・
nodes
・
・
= compactification of
(“Deligne-Mumford compactification” 1969)
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Recall:
Bers Theorem (Acta Math. 130. 1973 p103)
fiber space
s.t.
over a point
we have the fiber
Compactification
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The aim of this talk is to prove
Theorem The compactified fiber space
is the universal degenerating family, i.e.
for any fiber space (of genus
) with
non-constant moduli:
(at present
)
“pull back diagram”
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Symmetry of Riemann surface at finite or infinite palce
idea
Bers : Acta Math. 141 (1978)
Bers – Thurston classification of mapping classes
1. periodic (elliptic)
2. hyperbolic (“irreducible” not elliptic) (pseudo-Anosov)
3. parabolic “reduced” by
periodic
4. pseudo-hyperbolic (“reducible” not parabolic)
acts on
: periodic
has fixed points in
: parabolic
has fixed points at
of
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fixed point
fixed point
parabolic
periodic
cf.
hyperbolic
6
idea (bis)
Classification of degenerating family over
topological monodromy
Def
: pseudo-periodic
: periodic or
: parabolic
Theorem (M. & Montesinos 1991, 1992) Bull. AMS ’94
bijection
pseudo-periodic mapping
classes of negative twist
conjugation
: top. monodromy
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Bers constructed a fiber space (by simultaneous uniformization)
Fuchsian group
“Extended modular group”
acts on
: normalizer of
in
: id. on
“modular group”
“tautological fiber space”
acts on
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: periodic
●
●
quotient
singular fiber
whose topological
monodromy =
periodic
●
9
: parabolic
singular fiber whose
topological monodromy =
parablolic
●
quotient
“compactified”
●
“compactified”
10
Some details (well-known)
(geodesic) pants decomposition
closed geodesics
Fenchel – Nielsen Coordinates
“geodesic length”
“twisting angle”
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Two basic Lemmas (cf. Abikoff’s Lecture Note, LNM 820 pp 95-)
Lemma 1
a universal constant
s.t. geodesic simple loops
with
“short simple closed curves do not intersect”
long
short
long
short
Lemma 2
a universal constant
s.t. every Riemann surface
has a pants decomposition with curves
of length
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Compactification Process of
Given a set of infinite # of points
By the action of
, we may assume
infinite # of points
w.r.t. some pants decomposition (Fenchel – Nielsen coordinates)
Thus either
1.
convergent subsequence →
or
2.
(nodes)
13
To describe the second case, Bers introduced “Deformation space”
Riemann surface with nodes
free abelian group
generated by Dehn twists
“completion” of
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(“off axis” part)
(
)
15
Allowable map
(Bers)
deformation
allowable
“infinite cyclic covering”
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To obtain
of
.
, we must further make “quotient”
But we cannot “see” the action of
on
,
because the action of
is not well-defined on
Def
normalizer of
acts on
.
in
biholomorphically.
Teichmüller space of Riemann surfaces
of nodes
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small
-neighborhood of
in
where
preserves
In Bers classification:
• If
is parabolic, reduced by
,
is periodic in
• If
is pseudo-hyperbolic, reduced by
,
is hyperbolic in
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We can construct the compactification
Orbifold charts:
as an orbifold.
and
tautological fiber space (with smooth fiber) on
Type 1
but on
finite monodromy
Type 2
, we have singular fiber with
family of Riemann surfaces with nodes on
(cf. I. Kra 1990)
but on
, we have singular fiber with
pseudo-periodic monodromy
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Example
“parabolic”
180°
gives a full Dehn twist here
The corresponding singular fiber is
2
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fixed points of
quotient
2
MM Thm
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By Mat. – Montesinos Thm, there are no singular fibers other
than the above two types.
Thus the compactified fiber space
over
contains (essentially) all types of singular fibers over
.
The existence of the pull-back diagram in the Theorem
could be proved by the method of Imayoshi (1981).
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Thank you!
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