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

which is the universal coefficient theorem followed by the dual of the Hurewicz
isomorphism. Then ι N maps to the identity on Z.
(2) The homotopy groups of the infinite projective space are

Z for i = 2,
π i (P ∞ (C)) =
0 for i = 2.
Such spaces with exactly one nonzero homotopy group play a basic role in classify-
ing cohomology, and we say that such a space P ∞ (C) is a K(Z, 2)-space. For each
2
cohomology class c ∈ H (X, Z) there exists a map f : X → P ∞ (C) such that
c = f (ι ∞ ),and f is unique up to homotopy with this property.
∗
If X is finite dimensional, for example a manifold or if X is compact, then every
map f : X → P ∞ (C) factors by some inclusion P N (C) → P ∞ (C) giving a map
∗
g : X → P N (C) with the property that c = g (ι N ).
(3) The projective space P N (C) with homogeneous coordinates z 0 : ··· : z N has
an open covering of N +1 open coordinate domains given by z j = 0. On the domain
z j = 0, the coordinate function is
for k = j,0 ≤ k ≤ N.
z k ( j) = z k /z j
We have seen in (3.8) that the finite projective space P N (C) is a K¨ ahler manifold of
complex dimension N.
(4) The projective spaces have canonical line bundles L(N) → P N (C) where
L(∞) restricts to L(∞)|P N (C) = L(N). The space L(N) is the subspace of
P N (C) × C N+1 consisting of (z,λz) for λ ∈ C. For every topological C-line bun-
dle L → X over a space X there exists a map f : X → P ∞ (C) such that L and
∗
f (L(∞)) are isomorphic, and f is unique up to homotopy. If X is compact or finite
dimensional, then we can choose f : X → P N (C).
(5.2) First Chern Class of a Line Bundle. This is defined by observing that the set
of homotopy classes [X, P ∞ (C)] of mappings x → P ∞ (C) classify isomorphism
2
classes of topological complex line bundles L → X and also elements of H (X, Z)
2
by (5.1) (1). The cohomology class c 1 (L) ∈ H (X, Z) associated to the line bundle
L is called the first Chern class of L. The function
2
c 1 : {isomorphism classes of complex line bundles/X}→ H (X, Z)
is a bijection satisfying the multiplication property

c 1 (L ⊗ L ) = c 1 (L ) + c 1 (L ).
The higher Chern classes c j (E) of a complex vector bundle E have an axiomatic
characterization, due to Hirzebruch, based on the first Chern classes c 1 (L) of a com-
plex line bundle.
(5.3) Axioms for Chern Classes. The Chern class of a complex vector bundle E
ev 2i
over X is an element c(E) ∈ H (X, Z) = H (X, Z) satisfying the following
i≥0
properties: