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§7. Examples of Calabi–Yau Varieties from Toric Geometry 369

so that the Betti numbers satisfy the symmetry relation
i 2n−i
b i = dimH (X) = dimH (X) = b 2n−i .
DR DR
Serre duality is a nondegenerate pairing
i
H (X, L) × H n−i (X,ω X ⊗ L (−1)⊗ ) → k
for a line bundle L where ω X is the dualizing sheaf. For smooth manifolds, we have
n
ω X = . Hence the dimensions satisfy the symmetry
X
i
i
h (L) = dimH (X, L) = dimH n−i (X,ω X ⊗ L (−1)⊗ ) = h n−i (ω X ⊗ L (−1)⊗ ).
(6.10) Cohomology Properties of Calabi–Yau Manifolds. The fifth Calabi–Yau
condition that X is a K¨ ahler manifold with a trivial canonical line bundle ω X im-
plies that

0 for 0 < i < n,
0 i
H (X, ) =
X
k = C for i = 0, n.
This leads to vanishing of the E r terms. For example, the three dimensional diagram
becomes for the differential d 1 and ground field k
3 3 3
k = H (X, O X ) −−−−→ 0 −−−−→ 0 −−−−→ H (X, ) = k
X
2
2
1
2
0 −−−−→ H (X, ) −−−−→ H (X, ) −−−−→ 0
X X
1
1
2
1
0 −−−−→ H (X, ) −−−−→ H (X, ) −−−−→ 0
X X
0 0 3
k = H (X, O X ) −−−−→ 0 −−−−→ 0 −−−−→ H (X, ) = k
X
The corresponding nonzero Hodge numbers are either one or h 1,1 , h 2,1 = h 1,2 ,and
h 2,2 = h 1,1 by Poincar` e duality.
(6.11) Recommended Reading. We recommend that the reader consult now the
books of Cox and Katz [1999], Joyce [1998], and Voisin [1996].

§7. Examples of Calabi–Yau Varieties from ToricGeometry

The first examples of Calabi–Yau varieties were hypersurfaces or intersections of
hypersurfaces inprojective space. For these examples the concrete descriptionof the
canonical divisor is used. Projective spaces are special cases of weighted projective
spaces, and weighted projective spaces are special cases of toric varieties. In each
case hypersurfaces and complete intersections of hypersurfaces have a canonical di-
visor which is described interms the combinatorial data of the toric variety. Inmany
cases it is trivial,and this gives many more examples of Calabi–Yau varieties.