Complex Manifold Deformation Theory
Conjecture A
1 Distance of Word
Conjecture
Word has distance.
[Explanation]
1
Topological space E, B, F
Continuous map : EF
Homeomorphic with F -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 {Ua}aA is that the next is satisfied.
E = aAUa
Open sets family { Ua}aA 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 : 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 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= fAAO :HOH'O that is compatible with filter F is called sheaf from H to H'.
8
Deformation's morphism of Hodge structure : HA HAA (-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 disk D = { 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 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 limz0p(z)
Universal 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 dD(
(w), (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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