Complex Manifold Deformation Theory
Conjecture A
3 Uniqueness of Word
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 =
hijzi
j
Correspondence differential form w
w =
hijdzi
d
j
When dw = 0, h is called K
ller metric and w is called K
llerform.
Complex manifold that has K
ller metric K
ller manifold
2
n-dimensional compact K
ller manifold X
K
ller metric of X g
Correspondence K
ller form w
Local coordinate system z1, ..., zn
Ricci curvature of X R
= -
log det (g
)
Ricci form Ric(w) =
R
dz
d
A certain constant c
When Ric(w) = cw, g is called K
ller-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 K
ller manifold X
X satisfies c1(X) = 0 or c1(X) <0.
When K
ller form's cohomology class is fixed, K
ller-Einstein metric exists uniquely.
[Comment]
When word is expressed by compact K
ller manifold in adequate condition, word has
uniqueness.
[References]
Especially the next is important.
To be continued
Tokyo December 11, 2008
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