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§8. Line Bundles and Divisors: Picard and N´ eron–Severi Groups 373

2
The group Pic(X) is related to H by the exponential function.
(8.4) First Chern Class of Analytic/Algebraic Line Bundles. Consider the expo-
nential sequences
e
0 −−−−→ Z −−−−→ O Z −−−−→ O ∗ −−−−→ 1
X
where e( f ) = exp(2πif ). The boundary morphism in the cohomology exact se-
quence
c 1
1
2
2
1
∗
... →H (X, O X ) → H (X, O ) = Pic(X) → H (X, Z) → H (X, O X ) → ...
X
2
is the first Chern class c 1 :Pic(X) → H (X, Z) of line bundle classes. From the
exact sequence we see that this algebraic or analytic Chern class is an isomorphism
2
1
if H (X, O X ) = H (X, O X ) = 0.
(8.5) Definition. Let
2
0
Pic (X) = ker(c 1 :Pic(X) → H (X, Z))
contained in Pic(X). The N´ eron–Severi group of X is the quotient NS(X) =
0
Pic(X)/Pic (X).
With the intersection form we will give another description of the N´ eron–Severi
group NS(X) as a quotient of Pic(X) in (7.7).
In the algebraic case there is a purely algebraic first Chern class in ´ etale coho-
mology using the Kummer sequence instead of the exponential sequence.
(8.6) Degrees and Intersection Properties of Divisors. We consider the theory for
curves, surfaces, and threefolds.
(1) For a curve X, the deg : Div(X) → Z is a function which defines on the
quotient deg : Pic(X) = Div(X)/Div p (X) → Z by the first Chern class evaluated
on the top class [X] ∈ H 2 (X, Z) in homology
deg(ϕ) = c 1 (ϕ)[X]ordeg(D) = c 1 ((D))[X].
Itdefines a quotientmorphism on Pic(X) since deg(( f )) = 0, that is, the number of
zeros equals the number of poles of a function f .
(2) For a surface X, deg is replaced by the intersection pairing defined by the first
Chern class cup product and evaluated on the top class
L 1 · L 2 = c 1 (L 1 )c 1 (L 2 )[X]or D 1 · D 2 = c 1 (D 1 )c 1 (D 2 )[X].

The intersection pairings on surfaces are defined
Div(X) × Div(X)



Pic(X) × Pic(X) −−−−→ Z



2
2
H (X, Z) × H (X, Z)