Conjecture
Word has distance.
[Explanation]
1
Topological space E, B, F
Continuous map 
: E
F
: EF
Homeomorphic with F 
-1 (b) , b
B
-1 (b) , bB
Neighborhood of b U
B
B
Homeomorphic with U 
F 
-1 (U)
F -1 (U)
Homeomorphic map h : 
-1 (U) 
U 
F
-1 (U) U F
Objection to primary component p1 : U 
F 
U 
, h and p1 are fiber bundle in total
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
}aA
What E is covered by {Ua}a
A is that the next is satisfied.
A is that the next is satisfied.
E = 
a
AUa
aAUa
Open sets family { Ua}a
A is called open covering.
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
: VW
Map 
that satisfies the next is called analytic family of compact complex manifolds.
that satisfies the next is called analytic family of compact complex manifolds.
(i) 
is proper map.
is proper map.
(ii) 
is smooth holomorphic map.
is smooth holomorphic map.
(iii) For arbitrary point of w
W, fiber 
-1 (w) is always connected.
W, fiber -1 (w) is always connected.
When w0
W is fixed, Vw, w
W is called deformation of Vw0.
W is fixed, Vw, wW is called deformation of Vw0.
4
Complex manifold S
Weight w
Deformation of polar Z-Hodge structure H = (HZ, F, 
)
)
Point s0 
S
S
HZ = HZ (s0)
Fp = Fp(s0) 
= 
(s0)
= (s0)
Polar Z-Hodge structure (HZ, {Fp}, 
)
)
Period domain that is canonical by (HZ, { Fp}, 
) D
) D
compact relative of D 
Bilinear form over HZ, that is determined by
Q
Q
Monodromy expression of S's fundamental group 
(S, s0) 
: 
1 (S, s0) 
GZ =
(S, s0) : 1 (S, s0) GZ =
Aut(HZ, Q)
= Im 
= 
( 
1 (S, s0) )
: S 
\ D
is called period map.
= Im = ( 1 (S, s0) ) : S \ D is called period map.
5
Compact manifold M
Horizontal tangent bundle Th
Regular map 
: M 
: M
Horizontal d
is map that is from TM to Th(
)
is map that is from TM to Th()
Locally liftable 
| V : V 
D 
\ D
| 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 .
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
H'A
fo= fA
AO :HO
H'O that is compatible with filter F is called sheaf from H to H'.
AO :HOH'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)
: HA HAA (-w)
s
S
S
Fiber 
A(s)= 
A,s
A(s)= A,s
Weight w
that gives polar of w's A-Hodge structure at s is called polar of deformation
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}
C | |z|<1 }, D* = D\{0}
Universal covering of D* Upper half-plane of Poincaré H
Covering map H 
z 
exp(2
z) 
D*
z exp(2z) D*
Polarized VHS on D (H,S)
Fundamental group 
1(D*) 
Z
1(D*) Z
Generation element of the fundamental group HC
Action as monodromy to HC T
Period map adjoint with H p : H 
D
D
p ( z + 1 ) = Tp(z)
10
O module of deformation of Hodge structure H HO
D* = D\{0}
Period map p : D* 
\ D
\ D
Limit of p limz
0p(z)
0p(z)
Universal covering H
D
D
11
Period map 
Nilpotent orbit 
(w) : = exp(wN) 
(0)
(w) : = exp(wN) (0)
(Nilpotent orbit theorem)
(i) Nipotent orbit is horizontable map.
(ii) If Im w > 0 is enough large, 
(w) 
D.
(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 .
(w) )(Imw)Be-2Imw .
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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