Mathematics G4403. Modern Geometry
Spring 2016
Assignment 22
Due Monday, April 18, 2016
(1) Let Γ = {Hp : p ∈ S 2n+1 } be the connection on the principal U (1)-bundle
π : S 2n+1 → Pn (C) defined in Assignment 21 (2), and let ĝ be the Riemannian metric on Pn (C) defined in Assignment 21 (2). We now specialize
to the case
√ n = 1. Let C be oriented by teh volume form dx ∧ dy, where
z = x + −1y is the complex coordinate on C. We choose an orientation
on P1 (C) such that
h
z
1
,p
f : C → P1 (C), z 7→ p
1 + |z|2
1 + |z|2
is orientation preserving. Let ν ∈ Ω2 (P1 (C) be the volume form determined by this orientation and the Riemannian metric ĝ.√Let Ω = Dω ∈
√
−1Ω2 (S 3 ) be the curvature form of ω. Show that Ω = −1cπ ∗ ν, where
ν is the volume form defined by ĝ, and c is a real constant. Find c.
(2) Let E be a C ∞ complex vector bundle over a C ∞ manifold M , and let
∇ be a connection on E. The induced connection on End(E) = E ∗ ⊗ E
satisfies
∇X (φ(s)) = (∇X φ)(s) + φ(∇X s)
∞
for all φ ∈ C (M, End(E)), X ∈ X (M ), s ∈ C ∞ (M, E). The exterior
covariant derivative d∇ : Ωr (M, End(E)) → Ωr+1 (M, End(E)) satisfies
d∇ (α ⊗ φ) = dα ⊗ φ + (−1)r α ∧ ∇φ
for all α ∈ Ωk (M ), φ ∈ C ∞ (M, End(E)). The trace map End(E) → C,
where C = M × C is the product (trivial) complex line bundle over M , induces tr : Ωk (M, End(E)) → Ωk (M, C). Prove that if φ ∈ Ωk (M, End(E))
then
d(trφ) = tr(d∇ φ).
(3) Let (E, h) be a hermitian vector bundle over a C ∞ manifold M , and let ∇
be a unitary connection. Let F ∈ Ω2 (M, End(E)) be the curvature 2-form
of ∇. Use (2) and the Bianchi identity to show that
√−1 k
chk (E, ∇) := tr
F
2π
is a closed 2k form for any positive integer k ≤ dim M/2.
(4) Let (E, h) be a hermitian vector bundle over a C ∞ manifold M , and let ∇
be a unitary connection. We also use ∇ to denote the induced connection
on the dual vector bundle E ∗ , and on Λk E. The determinant line bundle
of E is defined to be det E := Λr E, where r = rankC E. Prove the following
identities of Chern forms.
(a) ck (E ∗ , ∇) = (−1)k ck (E, ∇).
(b) c1 (E, ∇) = c1 (det E, ∇).