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§6. Characterizations of Calabi–Yau Manifolds: First Examples 367

(6.5) Example (dimension two). Either a smooth quartic surface in P or the com-
plete intersection of a smooth quadric with a smooth cubic hypersurface in P 4 is a
Calabi–Yau in dimension two. The Euler number is 24. There are two new consider-
ations not seen in dimension one.
(1) These examples are special cases of K3 surfaces, for unlike smooth cubic curves
in P 2 giving all smooth curves of genus one, not every K3 surface is of this form.
(2) There are K3 surfaces over the complex numbers which are not algebraic sur-
faces. On the other hand, the Euler number is always 24.

These facts will be partly explained in the next two sections when the theory of
K3 surfaces is put in the general Enriques classification theory of surfaces.

(6.6) Example (threefolds). There are five cases of complete intersections of
smooth hypersurfaces in P 3+m in general position with the resulting variety is a
Calabi–Yau
(1) A quintic in P 4 , and the Euler number is −200.
(2) The intersection of a quartic and quadratic in P 5 , and the Euler number is −176.
The intersection of two cubics in P 5 , and the Euler number is −144.
(3) The intersection of a cubic and two quadrics in P 6 , and the Euler number is
−144.
(4) The intersection of four quadrics in P 7 , and the Euler number is −128.
(6.7) Remark. More generally, the m dimensional complete intersection X of k
smooth hypersurfaces of degree d 1 ,..., d k in P m+k = P is a variety with trivial
first Chern class zero if and only if d 1 +· · · + d k = m + k + 1. This follows from
the following exact sequence for the tangent sheaf
k

0 → T X → T P |X → O X (d i ) → 0
i=1
* + * +
giving the canonical sheaf K X = K P |X i d i = (O P (−m − k − 1)|X) i d i so
* +
that K X = O X −m − k − 1 + i d i .
Cohomology provides the first invariants of varieties and Calabi–Yau manifolds
in particular. For this we look closer at the Hodge to de Rham spectral sequence
considered before in (2.11), (3.11), and (3.12) especially in low dimensions.

(6.8) The Hodge to de Rham Spectral Sequence. The Betti numbers and Hodge
p,q i
numbers are related by b i = h for the cohomology H (X, C) has a
p+q=1
p
p
q
p
i
decreasing filtration F H (X, C) with H (X,& ) isomorphic to F H p+q (X, C)/
X
F p+1 H p+q (X, C). This is all related to a Hodge to de Rham spectral sequence where
the first differential
i, j j i i+1, j j i+1
d 1 : E = H (X,& ) → E = H (X,& )
1 X 1 X
is induced by the holomorphic differential