Complex Manifold Deformation Theory 1 Distance of Word

Complex Manifold Deformation Theory 

1 Distance of Word

TANAKA Akio  

Conjecture

Word has distance.

[Explanation]
1
Topological space     E, B, F
Continuous map     : EF
Homeomorphic withF     ^-1 (b) , bB    
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 }_aA
What E is covered by {U_a}_aA is that the next is satisfied.
E = _aAU_a
Open sets family { U_a}_aA is called open covering.
What covering is simply connected in space is called unversal covering.

3
Complex manifold     M
Point of M     Q
Normal tangent vector space     T_Q(M)
m+n dimensional complex manifold     V
m dimensional complex manifold      W
Holomorphic map     : VW
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 wW, fiber ^-1 (w) is always connected.
When w₀W is fixed, Vw, wW is called deformation of V_w0.

4
Complex manifold     S
Weight     w
Deformation of polar Z-Hodge structure     H = (H_Z, F, )
Point     s₀ S
H_Z = H_Z (s₀)
F_p = Fp(s₀)
=  (s₀)
Polar Z-Hodge structure     (H_Z, {F^p}, )
Period dmain that is canonical by (H_Z, { F^p}, )     D
compact relative of D    
Bilinear form over H_{Z, t}hat is determend by    Q
Monodromy expression of S's fundamental group (S, s₀)       : ₁ (S, s₀) G_Z = Aut(H_Z, Q)
 = Im =   ( ₁ (S, s₀) )
 : S \ D
  is called period map.

5
Compact manifold     M
Horizontal tangent bundle     T^h
Regular map      : M  
Horizontal     d is map that is from TM to T^h()
Locally liftable     | V : V D \ D

6
Subring of R      A
H = (H_A, F) that satisfies the next is called weight w's A-Hodge structure.
(i) H_A is finite generative A module.
(ii) For arbitrary p, q, there exists decomposition H_C= _p+q=wH^p,q that satisfies H^p,q = H^p,q .   H^p,q is complex conjugate for H^p,q .

7
A-Hodge's deformation over S      H = (H_A, F),  H' = (H'_A, F)   
Morphism of A module's local constant sheaf      f_A: H_A H'_A
f_o= f_AA_O : H_OH'_O that is compatible with filter F is called sheaf from H to H'.

8
Deformation's morphism of Hodge structure     : H_A  H_AA (-w)
sS
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 diskD = { z C | |z|<1 }, D* = D\{0}
Universal covering of D*     Upper half-plane of Poincaré      H
Covering map     H  z exp(2z) D*
Polarized VHS on D     (H,S)

Fundamental group     1(D*)   Z
Generation element of the fundamental group     H_C
Action as monodromy to H_{C     T
Period map adjoint with H     p : H D
p ( z + 1 ) = Tp(z)}

10
O module of deformation of Hodge structure H     H_O
D* = D\{0}
Period map     p : D* \ D
Limit of p   lim_z0p(z)
Unversal covering     HD

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 d_D((w), (w) )(Imw)^Be^-2Imw .
     d_D is invariant distance over D.

[Comment]
When word is expressed by open disk D, word has invariant distance in adequite condition(Im w > 0).
At that time, B is proper number of its word.

[Reference]
Distance Theory / Tokyo May 5, 2005 / Sekian Linguistic Field

Tokyo November 30, 2008

[Reference 2 / December 9, 2008]
Mirror Theory Group / Tokyo December 9, 2008 / Sekinan Linguistic Field

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