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368 19. Higher Dimensional Analogs of Elliptic Curves: Calabi–Yau Varieties

d d d d
0 −−−−→ O X −−−−→ & 1 −−−−→ ... −−−−→ & i −−−−→ ... .
X X
i, j
The spectral sequence E r is defined for all r ≥ 1 with differentials
i, j i+r, j−r+1
d r : E r → E r
such that H(E r , d r ) = E r+1 .
p,q p+1,q
In dimension 1, we have the differential d 1 : E → E
1 1
1
0
1
H (X, O X ) −−−−→ H (X,& )
X
1
0
1
H (X, O X ) −−−−→ H (X,& )
X
p,q p+1,q
In dimension 2, we have the differential d 1 : E → E
1 1
2
2
2
1
2
H (X, O X ) −−−−→ H (X,& ) −−−−→ H (X,& )
X X
1
1
2
1
1
H (X, O X ) −−−−→ H (X,& ) −−−−→ H (X,& )
X X
1
0
2
0
0
H (X, O X ) −−−−→ H (X,& ) −−−−→ H (X,& ).
X X
p,q p+1,q
In dimension 3, we have the differential d 1 : E → E
1 1
3
2
3
1
3
3
3
H (X, O X ) −−−−→ H (X,& ) −−−−→ H (X,& ) −−−−→ H (X,& )
X X X
2
2
2
2
1
2
3
H (X, O X ) −−−−→ H (X,& ) −−−−→ H (X,& ) −−−−→ H (X,& )
X X X
3
2
1
1
1
1
1
H (X, O X ) −−−−→ H (X,& ) −−−−→ H (X,& ) −−−−→ H (X,& )
X X X
0
0
0
3
2
0
1
H (X, O X ) −−−−→ H (X,& ) −−−−→ H (X,& ) −−−−→ H (X,& ).
X X X
The differentials d r with r ≥ 1 are all zero for smooth algebraic varieties and
K¨ ahler manifolds.
(6.9) Poincar´ e Dualityand Serre Duality. For a complex manifold X of complex
dimension n we have a nondegenerate pairing
H i (X) × H 2n−i (X) → C
DR DR