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

(7.1) Projective Spaces. The first examples of Calabi–Yau varieties in dimension m
were the complete intersections of multiple degree (d 1 ,..., d k ) in projective space
P m+k satisfying

(CY)(d 1 ,..., d k ; m) : m + k + 1 = d 1 +· · · + d k

which is the condition for a trivial canonical bundle. When m = 3, for example, we
have the following solutions of 4 + k = d 1 +· · · + d k with d i > 1:

5 = 5, 6 = 4 + 2or6 = 3 + 3, and 7 = 3 + 2 + 2
In looking for further three dimensional examples, we consider complete inter-
sections in weighted projective spaces.
(7.2) Weighted Projective Spaces. A weighted projective space P N (w) is a gener-
alization of P N . both are quotients of C N+1 −{0} by an action of C = C−{0}.The
∗
weights w are sequences of natural numbers w = (w(0), ...,w(N)) ∈ N N+1 and
∗
the action of λ ∈ C on (z 0 ), ..., z N ) ∈ C N+1 −{0} is given by the formula
λ · (z 0 ,..., z n ) = (λ w(0) z 0 ,...,λ w(N) z N ).
We assume that the greatest common divisor of the w(i) is 1.

For each subset S ⊂{w(0), ...,w(N)} we denote by q(S) the greatest common
divisor of the w(i) with i ∈ S. Let H(S) denote the subset of all (z j ) ∈ P N (w) with
z i = 0 for i /∈ S. The points in H(S) are cyclic quotient singularities for the group
Z/q(S)Z.
A general reference on weighted projective spaces is I. Dolgachev [1982, SLN
956].
The equations of hypersurfaces in the weighted projective space P N (w) of degree
d are given by polynomial equations f (z 0 ,..., z n ) = 0 where

w(0) w(N) d
f (λ z 0 ,...,λ z N ) = λ f (z 0 ,..., z N ).
(7.3) Complete Intersections in Weighted Projective Spaces. The complete inter-
sections of multiple degree (d 1 ,..., d k ) in the weighted projective space P m+k (w)
with trivial canonical bundle are those satisfying the following condition
(CY)(d 1 ,..., d k ; w) : w(0) +· · · + w(m + k) = d 1 +· · · + d k .

This condition reduces to (CY)(d 1 ,..., d k ) in (7.1) for a projective space P m+k .

The existence of singularities in a weighted projective space, which were not
present in the standard projective space, leads to examples with these quotient sin-
gularities. Of special interest are the hypersurfaces transverse to the singularities.
There is a complete classification of Calabi–Yau varieties arising from transverse
hypersurfaces in P 4 (w), see A. Klemm and R. Schimmrigk [1994] and M. Kreuzer
and H. Skarke [1992], where there are 7555 cases.