First Chern Form on $\mathbb{CP}^1$¶

A complex line bundle is characterized by the curvature of a unitary connection. On an affine chart of $\mathbb{CP}^1$, use the complex coordinate

$$u=r e^{2\pi i\theta},\qquad \bar u=r e^{-2\pi i\theta}.$$

We compute a Fubini--Study connection, its curvature, and the normalized first Chern form.

Polar chart and connection one-form¶

The chart variables are $(r,\theta)$ with $r\ge0$ and $0\le\theta<1$. The local connection is

$$\omega=\frac{\bar u\,du}{1+u\bar u}.$$

declare symbol r, θ : MathValue

def params : Vector MathValue := [| r, θ |]
def u : MathValue := r * e ^ (2 * π * i * θ)
def ū : MathValue := r * e ^ ((-2) * π * i * θ)

def d (X : MathValue) : DiffForm MathValue :=
  !(flip ∂/∂) params X

def ω : DiffForm MathValue := ū * d u / '(1 + u * ū)

ω

$\begin{pmatrix} \frac{r}{r^{2} + 1} \\ \frac{2 i r^{2} π}{r^{2} + 1}\\ \end{pmatrix}$

Curvature two-form¶

For a line bundle the connection is Abelian, so $\omega\wedge\omega=0$ and $\Omega=d\omega$. We display its antisymmetric component matrix explicitly.

def ωr : MathValue := ω_1
def ωθ : MathValue := ω_2
def Ωrθ : MathValue :=
  (∂/∂ ωθ r - ∂/∂ ωr θ) / 2
def Ω : DiffForm MathValue :=
  [| [| 0, Ωrθ |], [| -Ωrθ, 0 |] |]

Ω

$\begin{pmatrix} 0 & \frac{2 i r π}{r^{4} + 2 r^{2} + 1} \\ \frac{-2 i r π}{r^{4} + 2 r^{2} + 1} & 0 \\ \end{pmatrix}$

First Chern form¶

With the orientation and connection convention used here,

$$c_1=\frac{\Omega}{-2\pi i}.$$

def c1Form : DiffForm MathValue := Ω / ((-2) * π * i)

c1Form

$\begin{pmatrix} 0 & \frac{-r}{r^{4} + 2 r^{2} + 1} \\ \frac{r}{r^{4} + 2 r^{2} + 1} & 0 \\ \end{pmatrix}$

The full antisymmetric form contributes twice its upper component, so its oriented density is

$$-\frac{2r}{(1+r^2)^2}\,dr\wedge d\theta.$$

Integrating over $r\in[0,\infty)$ and $\theta\in[0,1)$ gives

$$\int_0^\infty\frac{-2r}{(1+r^2)^2}\,dr=-1.$$

Thus this convention describes the degree $-1$ line bundle. Reversing the orientation or using the dual connection reverses the Chern number.

First Chern Form on $\mathbb{CP}^1$¶

Polar chart and connection one-form¶

Curvature two-form¶

First Chern form¶

Links