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§5. Projective Spaces, Characteristic Classes, and Curvature 365

(5.9) Curvature Forms for the K¨ ahler Case. For a complex vector bundle E →
X over a K¨ ahler manifold X, the curvature form is given locally by a matrix of the
form

1,1
& = & i, j,k, dz k ∧ dz ∈ M N (A (X)).
k,
(5.10) First Chern Class. Returning to the calculation in (5.5), we see that c 1 (&) =
e 1 (&) = Tr(&). This trace is an other well known curvature form, the Ricci curvature
form,
1,1
Ric = & j, j,k, dz k ∧ dz ∈ A (X).
j,k,
In particular, the vanishing of the Ricci curvature form implies that the first Chern
1
class is trivial. If & = &, then we have also the following formula for the total
2πi
Chern class
2

c(X) = 1 + c j (X) = det(1 + & ) = 1 + tr(& ) + tr(& ∧ & − 2(tr(& )) ) + ....
j
In each case the conjugation invariance leads to intrinsic quantities independent
of framings. Now we consider some special features of line bundles and their first
Chern class.
(5.11) Remark. The group of line bundles up to isomorphism, with multiplica-
ˇ
tion given by tensor product, can be described as the Cech cohomology group
ˇ
ˇ 1
∗
H (X, O ). Here, a Cech cocycle
X
1
∗
g = (g a,b ) ∈ Z (U, O )
ˇ
X
defines a line bundle L on X by gluing trivial bundles on U a where U = (U a ) with
the invertible g a,b on the intersections U a ∩U b .For g a,b = g −1 we see that the lifting
b,a
1 1
σ a,b = log(g a,b ) ∈ C (X, O X ) has a coboundary
2πi
1 * + 2
ˇ
(δσ) a,b,c = log(g b,c ) + log(g c,a ) + log(g a,b ) ∈ Z (U, Z).
2πi
ˇ
This is a Cech cocycle for c 1 (L). We return to this subject in (7.4).
(5.12) Remark. For the local calculation in (4.8), we saw that a Hermitian metric h
on a complex analytic line bundle leads to a holomorphic connection ∇ h with local

connection matrix ω a = (d h a )h −1 where h a is the value of h on the frame over
a
U a as in the previous paragraph (5.11). The curvature of a line bundle is the exterior
derivative of the connection form

& a = dω a = d (h a ) ∧ d (h −1 ) = d d log(h a ).
a
Hence the first Chern class is given by the following de Rham cohomology class:
1

c 1 (L, h) = d d log(h a ).
2πi