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

(11.4) Signature and Intersection Form. The intersection form on H 2 (X, Z) or its
2
dual the cup product form on H (X, Z) is integral on this twenty-two dimensional
lattice. Since K X = 0, by (9.4) the signature

σ = (1/3)(K X · K X − 2e(X)) = (−2/3)(e(X)) = (−2/3) · 24 =−16

There is just one possibility for such an integral form, and it is

0 1
3 ⊕ (−2)E 8 = (3, 19)
1 0

0 1
where denotes the rank two hyperbolic form and E 8 is the unique even
1 0
form of rank 8.
(11.5) Remark. We denote the symmetry group by = (3, 19) = Aut( (3, 19))
2
⊂ SO(3, 19). Consider H = H = H 2 the lattice of rank 22 with the cup product
or intersection form respectively. Then the set Isom( (3, 19), H) is a right principal
homogeneous -set with action given by right composition.
2 2
(11.6) Real Structure on H (X,C). Consider the Hodge structure on H (X, C) =
2
2
H 2,0 ⊕ H 1,1 ⊕ H 0,2 . Complex conjugation c : H (X, C) → H (X, C) satisfies
2
c = id and it interchanges the two summands H 2,0 and H 0,2 . The intersection form
restricts to a positive definite form on the two dimensional subspace
2
2
P(X) = (H 2,0 ⊕ H 0,2 ) ∩ H (X, R) ⊂ H (X, R).
1,1 1,1 2
This implies that the restriction of the intersection form to H = H ∩ H (X, R)
R
1,1
has signature (1, 19).In H ,wedenoteby V (X) the subspace of elements of
R
1,1
strictly positive norm. Since the signature is (1, 19), V (X), the subset of all x ∈ H
R
−
with (x|x)> 0, has two disjoint components V (X) = V (X) ∪ V (X) where
+
V (X) denotes the component with K¨ ahler class.
+
In (7.2) and (7.8) we introduced the general theory of the Picard lattice of line
2
bundles and the N´ eron–Severi group. The kernel of c 1 :Pic(X) → H (X) was
0
0
denoted Pic (X) and the quotient Pic(X)/Pic (X) = NS(X) is the N´ eron–Severi
0
1
1
group. Since H (X, O X ) maps onto Pic (X) and H (X, O X ) = 0foraK3surface,
we see that the natural map Pic(X) → NS(X) is an isomorphism. We can in fact
bound the size of the N´ eron–Severi group using the real cohomology.
(11.7) Picard Lattice or N´ eron–Severi Group. The first Chern class monomor-
2
phism c 1 :Pic(X) → H (X) restricts to an isomorphism defined c 1 :Pic(X) →
2
2
H (X, Z)∩ H 1,1 ⊂ H (X, Z)∩ H 1,1 . The assertion that c 1 is an isomorphism in this
R
case is a theorem of Lefschetz. The Picard number or rank ρ(X) of Pic(X) satisfies
ρ(X) ≤ 20 and the Hodge algebraic index theorem (8.6) says that the signature is
(1,ρ(X) − 1).