Complex Manifold Deformation Theory
TANAKA Akio
Topological space E , B , F
Homeomorphic with F
-1 ( b ) , b
B
Homeomorphic with U
F
-1 ( U )
Homeomorphic map h :
-1 ( U )
U
F
Objection to primary component p 1 : U
F
U
, h and p 1 are fiber bundle in total
space S, base space B , fiber F and projection
.
Family that consists of E' s open sets { U
} a
A
What E is covered by { U a } a
A is that the next is satisfied.
Open sets family { U a } a
A is called open covering.
What covering is simply connected in space is called universal covering.
Normal tangent vector space T Q ( M )
m+n dimensional complex manifold V
m dimensional complex manifold W
Map
that satisfies the next is called analytic family of compact complex manifolds.
(ii)
is smooth holomorphic map.
(iii) For arbitrary point of w
W , fiber
-1 ( w) is always connected.
When w 0
W is fixed, Vw , w
W is called deformation of V w 0 .
Deformation of polar Z -Hodge structure H = ( H Z , F ,
)
Polar Z-Hodge structure ( H Z , { F p },
)
Period domain that is canonical by ( H Z , { F p },
) D
Bilinear form over H Z, t hat is determined by
Q
Monodromy expression of S' s fundamental group
( S , s 0 )
:
1 ( S , s 0 )
G Z = Aut( H Z ,
Q )
= Im
=
(
1 ( S , s 0 ) )
: S
\ D
is called period map .
Horizontal tangent bundle T h
Horizontal d
is map that is from TM to T h (
)
Locally liftable
| V : V
D
\ D
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=w H p,q that satisfies H p,q = H p,q .
H p,q is complex conjugate for H p,q .
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 A
A O : H O
H ' O that is compatible with filter F is called sheaf from H to H '.
Deformation's morphism of Hodge structure
: H A
H A
A (- w )
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 .
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 *
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
O module of deformation of Hodge structure H H O
Nilpotent orbit
( w ) : = exp( w N)
(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) B e -2
Im w .
d D is invariant distance over D .
When word is expressed by open disk D , word has invariant distance in adequate
At that time, B is proper number of its word.
Distance Theory / Tokyo May 5, 2005 / Sekinan Linguistic Field
Sekinan Research Field of language
[Reference 2 / December 9, 2008]
Mirror Theory Group / Tokyo December 9, 2008 / Sekinan Linguistic Field
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