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§5. Projective Spaces, Characteristic Classes, and Curvature 361

preserving a Hermitian product on E. Then the holonomy group, which is con-
tained in GL(N, C), the complex linear group for N = dimE. It is in the subgroup
U(N) ⊂ GL(N, C) in the case of a K¨ ahler manifold where the complex analytic
tangent bundle is T (X) (1,0) .
The holonomy group, being a closed subgroup of GL(N, C), has a Lie alge-
bra which is a subalgebra of LieGL(N, C) = M N (C). The restriction of exp :
M N (C) → GL(N, C) to this Lie algebra maps into the holonomy group. Ambrose
and Singer describe this sub-Lie algebra with curvature.
(4.13) Curvature Generates the Lie Algebra of the Holonomy Group. An early
reference in this direction is the paper of Ambrose and Singer, TAMS, 75 (1953).

(4.14) Remark. The connection associated with a metric has the property that paral-
lel transport preserves the metric. A Hermitian metric is K¨ ahler if and only if parallel
transport commutes with multiplication by i.

(4.15) SU(N)-holonomy and Ricci Curvature. A fundamental property of

exp : M N (C) → GL(N, C)
is that exp(tr(A)) = det(exp(A)). In particular, exp : su(N) → SU(N) is a local
surjection which restricts from exp : u(N) → U(N).

Here is where the Calabi–Yau theory starts from the complex differential geom-
etry perspective in the next sections.

§5. Projective Spaces, Characteristic Classes, and Curvature

For additional details to this sketch of the theory of Chern classes see the last chapters
of Husem¨ oller, Fibre Bundles. The theory of characteristic classes begins with the
first Chern class of a line bundle. This is a topological theory starting with complex
projective space, and the fact that many considerations in complex geometry, K¨ ahler
geometry, and algebraic geometry start with projective space suggests that Chern
classes are basic to many areas of geometry. This is the case.

(5.1) Remark. The finite and infinite complex projective spaces P N (C) ⊂ P ∞ (C)
have four aspects.
(1) The second cohomology group is infinite cyclic with a canonical generator
2
H (P N (C), Z) = Zι N such that ι ∞ restricts to ι N for all N. For the reader with a
little background in topology, this can be seen from the isomorphism
2
H (P N (C), Z) = Hom(H 2 (P N (C), Z), Z)
→ Hom(π 2 (P N (Z)), Z) = Hom(Z, Z)