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Conjecture A
1 Distance of Word
Conjecture
Word has distance.
[Explanation]
1
Topological space E, B, F
Continuous map
: E
F
Homeomorphic with F
-1 (b) , b
B
Neighborhood of b U
B
Homeomorphic with U
F
-1 (U)
Homeomorphic map h :
-1 (U)
U
F
Objection to primary component p1 : U
F
U
, h and p1 are fiber bundle in total
space S, base space B, fiber F and projection
.
2
Topological space E
Family that consists of E's open sets {U
}a
A
What E is covered by {Ua}a
A is that the next is satisfied.
E =
a
AUa
Open sets family { Ua}a
A is called open covering.
What covering is simply connected in space is called universal covering.
3
Complex manifold M
Point of M Q
Normal tangent vector space TQ(M)
m+n dimensional complex manifold V
m dimensional complex manifold W
Holomorphic map
: V
W
Map
that satisfies the next is called analytic family of compact complex manifolds.
(i)
is proper map.
(ii)
is smooth holomorphic map.
(iii) For arbitrary point of w
W, fiber
-1 (w) is always connected.
When w0
W is fixed, Vw, w
W is called deformation of Vw0.
4
Complex manifold S
Weight w
Deformation of polar Z-Hodge structure H = (HZ, F,
)
Point s0
S
HZ = HZ (s0)
Fp = Fp(s0)
=
(s0)
Polar Z-Hodge structure (HZ, {Fp},
)
Period domain that is canonical by (HZ, { Fp},
) D
compact relative of D
Bilinear form over HZ, that is determined by
Q
Monodromy expression of S's fundamental group
(S, s0)
:
1 (S, s0)
GZ = Aut(HZ,
Q)
= Im
=
(
1 (S, s0) )
: S
\ D
is called period map.
5
Compact manifold M
Horizontal tangent bundle Th
Regular map
: M
Horizontal d
is map that is from TM to Th(
)
Locally liftable
| V : V
D
\ D
6
Subring of R A
H = (HA, F) that satisfies the next is called weight w's A-Hodge structure.
(i) HA is finite generative A module.
(ii) For arbitrary p, q, there exists decomposition HC=
p+q=wHp,q that satisfies Hp,q = Hp,q .
Hp,q is complex conjugate for Hp,q .
7
A-Hodge's deformation over S H = (HA, F), H' = (H'A, F)
Morphism of A module's local constant sheaf fA: HA
H'A
fo= fA
AO :HO
H'O that is compatible with filter F is called sheaf from H to H'.
8
Deformation's morphism of Hodge structure
: HA
HA
A (-w)
s
S
Fiber
A(s)=
A,s
Weight w
that gives polar of w's A-Hodge structure at s is called polar of deformation of
w's A-Hodge structure's deformation.
Hodge structure that is associated with polar is called polarized VHS.
9
Open disk D = { z
C | |z|<1 }, D* = D\{0}
Universal covering of D* Upper half-plane of Poincaré H
Covering map H
z
exp(2
z)
D*
Polarized VHS on D (H,S)
Fundamental group
1(D*)
Z
Generation element of the fundamental group HC
Action as monodromy to HC T
Period map adjoint with H p : H
D
p ( z + 1 ) = Tp(z)
10
O module of deformation of Hodge structure H HO
D* = D\{0}
Period map p : D*
\ D
Limit of p limz
0p(z)
Universal covering H
D
11
Period map
Nilpotent orbit
(w) : = exp(wN)
(0)
(Nilpotent orbit theorem)
(i) Nipotent orbit is horizontable map.
(ii) If Im w > 0 is enough large,
(w)
D.
(iii) If Im w > 0 is enough large, there exists non-negative constant B that satisfies dD(
(w),
(w) )
(Imw)Be-2
Imw .
dD is invariant distance over D.
[Comment]
When word is expressed by open disk D, word has invariant distance in adequate
condition(Im w > 0).
At that time, B is proper number of its word.
[Reference]
Tokyo November 30, 2008
[Reference 2 / December 9, 2008]
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