Nonlinear Equations - UFRJ

[SEC. 4.5: PROJECTIVE SPACE 51
Moreover, by splitting T p E = ker Dπ ⊥ ⊕ ker Dπ and noticing that
F x = ker Dπ(p),
[ ]
Dπ(p) 0
DΦ =
.
? DΦ |Fx (p)
Therefore
det DΦ |Fx det DΦ −1 = det
(
Dπ −1
| ker Dπ ⊥ )
= (det DπDπ ∗ ) −1/2 .
When the fiber bundle is complex, we obtain a similar formula by
assimilating C n to R 2n :
Theorem 4.9 (co-area formula). Let (E, B, π, F ) be a complex smooth
fiber bundle. Assume that B is finite dimensional. Let f : E → R ≥0
be measurable. Then whenever the left integral exists,
∫
∫ ∫
f(p)dE(p) = dB(x) (det Dπ(p)Dπ(p) ∗ ) −1 f(p)dE x (p).
E
B
E x
with E x = π −1 (x).
4.5 Projective space
Complex projective space P n is the quotient of C n+1 \ {0} by the
multiplicative group C × . This means that the elements of P n are
complex ‘lines’ of the form
(x 0 : · · · : x n ) = {(λx 0 , λx 1 , · · · , λx n ) : 0 ≠ λ ∈ C} .
It is possible to define local charts at (p 0 : · · · : p n ) : p ⊥ ⊂ C n+1 → P n
by sending x into (p 0 + x 0 : · · · : p n + x n ).
There is a canonical way to define a metric in P n , in such a way
that for ‖p‖ = 1, the chart x ↦→ p + x is a local isometry at x = 0.
Define the Fubini-Study differential form by
√ −1
ω z =
2 ∂ ¯∂ log ‖z‖ 2 . (4.5)