Show the complex one-form defines a connection

Question

Let $E$ be the complex line bundle for $\mathbb {CP}^1$, $E=\{(L,v)\mid L\in Gr_1(\mathbb C^2),v\in L\}$. I know that this with zero section removed is diffeomorphic to $\mathbb C^2-\{0\}$. Now I want to show a complex valued $1$-form on $\mathbb C^2-\{0\}$, $\theta=\frac{\bar z_0 \, dz_0+\bar z_1 \, dz_1}{|z_0|^2+|z_1|^2}$, defines a connection on E. I'm completely stuck on this. In particular, even though I know the definition of connection on $E$, and the diffeomorphism between $E$ and $\mathbb C^2-\{0\}$ is given by project on second factor, I'm still confused about how to transfer this definition of one-form to a map on vector field and sections on E, where I can apply the usual definition of connection. Any help is appreciated.

Answer 1

This is the canonical (unique) connection on $E$ compatible with both its structure as a holomorphic line bundle and the standard hermitian structure on $E$ inherited from the hermitian inner product on $\Bbb C^2$. In particular, if $s(z)=Z=(1,z)$ is the obvious holomorphic section of $E$ over $\Bbb P^1-\{[0,1]\}$, you should check that $$\nabla_V Z = (s^*\theta)(V) Z$$ is the projection of $dZ(V)$ onto the line spanned by $Z$, i.e., the fiber of $E$ over $(1,z)$. (Think of this as generalizing the connection on a submanifold of $\Bbb R^n$ defined by projecting the directional derivative of a vector field onto the tangent space of the submanifold.) For a general section $s$ of $E$, you write $s(z) = \lambda(z)Z$ and set $\nabla_V s = V(\lambda)Z + \lambda\nabla_V Z$.