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

(3) For a threefold X the deg is replaced by a trilinear form on Pic(X)

L 1 · L 2 · L 3 = c 1 (L 1 )c 1 (L 2 )c 1 (L 3 )[X]
for line bundles, and
D 1 · D 2 · D 3 = c 1 (D 1 )c 1 (D 2 )c 1 (D 3 )[X]
for divisors.
Returning to the N´ eron–Severi group NS(X), we consider the following relation
between divisors.

(8.7) Definition. Two divisors D and D are numerically equivalent provided D ·
E = D · E for all divisors E on a surface X.

(8.8) Proposition. Let X be a surface such that the intersection form on Pic(X) is
nondegenerate. Then the N´ eron–Severi group NS(X) is the quotient Div(X)/Div n (X)
where Div n (X) is the subgroup of Div(X) consisting of divisors numerically equiv-
alent to zero.

By passing to the quotient, the intersection form D · D is defined on NS(X)
with values in the integers.
Since every divisor of a function is numerically equivalent to zero, that is,
Div p (X) ⊂ Div n (X), we have a quotient mapping from the divisor classes to
0
NS(X) = Pic(X)/Pic (X) = Div(X)/Div n (X) preserving the intersection form.
(8.9) Remark. The group NS(X) is finitely generated with rank ρ(X) which is
called the Picard number of the surface X. The intersection form extends to a bi-
linear pairing on the extension of scalars Q ⊗ NS(X) and R ⊗ NS(X). These are
vector spaces of dimension equal the Picard number ρ(X).

§9. Numerical Invariants of Surfaces

In this section we consider the invariants used in the Enriques classification of sur-
faces with a special emphasis on the surfaces satisfying the Calabi–Yau property.
This classification begins with cohomological invariants which extend the concept
of the genus of a curve to surfaces. Let k denote the algebraically closed ground
field.
(9.1) The Hodge to de Rham Spectral Sequence. It has the form
p,q q p
E = H (X,& )
1 X
with differentials induced by the complex of holomorphic differential forms
d d d d
0 −−−−→ O X −−−−→ & 1 −−−−→· · · −−−−→ & i −−−−→· · · .
X X
p,q p,q p+r,q−r+1
The terms E r and the differentials d r : E r → E r are defined for all
r ≥ 1, and H(E r , d r ) = E r+1 . We compare X of dimension 1 and X of dimension
2.