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§5. Projective Spaces, Characteristic Classes, and Curvature 363

(1) The Chern class is a sum c(E) = 1 + c 1 (E) +· · · + c n (E) where c 0 (E) = 1,
2i
c i (E) = 0 for i > dim(E) and c i (E) ∈ H (X, Z).
(2) If f : X → X is a map, and if E → X is a complex vector bundle over X, then

ev
c( f (E)) = f (c(E)) in H (X , Z) where f (E) is the induced bundle on X .
∗
∗

∗

(3) The Chern class of the Whitney sum E ⊕ E is the cup product

c(E ⊕ E ) = c(E )c(E )
ev
in H (X, Z).
(4) For a line bundle L, the Chern class c(L) = 1 + c 1 (L) where c 1 (L) is defined in
(5.2).
(5.4) Remark. As Grothendieck observed in algebraic geometry, the existence and
uniqueness of the Chern classes E → X can be established by working with the
related bundle P(E) → X of projective spaces and the standard line bundle L E →
P(E) reducing to the canonical line bundle on each fibre. The class c = c 1 (L E ) is
ev
ev
ev
2
in H (P(E), Z) and 1, c, c ,..., c n−1 is a basis H (P(E), Z) as a free H (X, Z)
ev
ev
module under the cup product preserving morphism H (X, Z) → H (P(E), Z)
induced by the projection P(E) → X. The Chern classes c i (E) are the coefficients
of the equation
n
n
c − c 1 (E)c n−1 +· · · + (−1) n−1 c n−1 (E)c + (−1) c n (E) = 0.
For further details, see Fibre Bundles, pp. 249–252.
In chapter 19 of Fibre Bundles we described how to define the Chern classes of
a complex vector bundle using a connection and its curvature form. We sketch this
theory and extend it to holomorphic connections on complex bundles on complex
manifolds.
(5.5) ElementarySymmetric Functions. The elementary symmetric functions

x
σ q (x 1 ,..., x n ) = i(1)<···<i(q) i(1) ... x i(q) are also encoded in the expression
q
Q x (t) = (1 + x j t) = σ q (x 1 ,..., x n )t .
1≤ j≤n 0≤q≤n
This leads to other polynomials e q which as polynomials of the symmetric function
σ q (x 1 ,..., x n ) have the form
d q
−t log Q x (t) = e q (σ 1 ,...,σ n )(−t) .
dt
q≥1
The ring of symmetric functions is the subalgebra of k[x 1 ,..., x n ] invariant under
the action of the symmetric group on the variables x 1 ,..., x n . It is itself a polynomial
ring on σ 1 ,...,σ n and it contains the functions e 1 ,..., e n .If k is of characteristic
zero, then k[σ 1 ,...,σ n ] = k[e 1 ,..., e n ].
Let k be a field of characteristic zero in the next parts.