Friday 29 March 2013

Complex Manifold Deformation Theory 3 Uniqueness of Word


Complex Manifold Deformation Theory 


3 Uniqueness of Word


TANAKA Akio  



Conjecture
Word has uniqueness.

[Explanation]
1
Smooth manifold     M
Hermitian metric of M     h    
Tangent bundle of      TM    
positive definite Hermitian inner product over TM      h
Local coordinate system     z1, ..., zn 

Hermitian symmetric positive definite function's matrix     hij    

h = hijzij
Correspondence differential form     w
w = hijdzidj
When dw = 0, h is called Kller metric and w is called Kllerform.
Complex manifold that has Kller metric     Kller manifold
2
n-dimensional compact Kller manifold     X
Kller metric of X     g

Correspondence Kller form     w
Local coordinate system z1, ..., zn 
Ricci curvature of X     R = - log det (g)
Ricci form    Ric(w) = Rdzd
A certain constant     c
When Ric(w) = cw, g is called Kller-Einstein metric.
When c = 0, c1(X) = 0
When c = -1, Linear bundle nTX 's cobundle is ample. The situation is briefly abbreviate expressed by c1(X) <0.
3
Compact Kller manifold     X
X satisfies c1(X) = 0 or c1(X) <0.
When Kller form's cohomology class is fixed, Kller-Einstein metric exists uniquely.


[Comment]
When word is expressed by compact Kller manifold in adequate condition, word has uniqueness.

[References]
Especially the next is important.
To be continued
Tokyo December 11, 2008
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