I'm trying to compute the Chern-class of the bundle $$\gamma = \{(c,\ell): c \in \ell \} \subseteq \mathbb{C}^2 \times \mathbb{C}P^1$$ over $\mathbb{C}P^1$. I'm running into a problem defining an affine connection on this bundle.
Any tips? So far I've tried to use the bundle $\mathbb{C}^2 \times \mathbb{C}P^1$. I don't see a natural way to take a derivative of a section of this bundle over $\mathbb{C}P^1$.
And yes, this computation is easy via topological argument, but I'm interested in the Chern-Weil computation.
$\newcommand{\dd}{\partial}\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$The total space of the tautological bundle is contained in the rank two trivial bundle $\Cpx\Proj^{1} \times \Cpx^{2}$, and the standard Hermitian structure on $\Cpx^{2}$ induces an Hermitian metric $h$ in $\gamma$, given in an affine chart by $$ h(z) = 1 + |z|^{2}, $$ the magnitude-squared of the local holomorphic section $(z, 1)$.
The Chern connection form is $$ \dd \log h = \frac{\bar{z}\, dz}{1 + |z|^{2}}, $$ and the curvature, which represents $2\pi c_{1}(\gamma)$, is $$ -i\, \dd \bar{\dd} \log h = -i\frac{dz \wedge d\bar{z}}{(1 + |z|^{2})^{2}}. $$ Integrating over the (dense) affine chart $\Cpx$ in polar coordinates shows the total curvature is $-2\pi$, which shows $c_{1}(\gamma)$ is the negative generator of second cohomology.