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.


1 Answer 1


$\newcommand{\dd}{\partial}\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$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.

  • $\begingroup$ Sorry, I'm not that familiar with complex geometry. Could you clarify a few things? So I'm happy with the Hermetian metric $h$ on $\gamma$. That is in the fiber over $[z:1]$ $h$ is just the standard inner product in $\mathbb{C}^2$ on points on the line through $(z,1)$. But how does this give us a (real) connection for the bundle? I'm unfamiliar with connections on general vector bundles, and I don't know how to construct one from that. I'm assuming some sort of metric compatibility? $\endgroup$ Dec 3, 2016 at 4:34
  • $\begingroup$ You're exactly right: An Hermitian metric in a holomorphic vector bundle has a unique compatible connection. In a holomorphic line bundle, an Hermitian metric is (effectively) a smooth, positive, real-valued function $h$ in each local chart; the associated connection form is $\theta = h^{-1}\, \dd h = \dd \log h$, and the curvature form is$$\Theta = i\, \bar{\dd}\theta = -i\, \dd \bar{\dd} \log h.$$ $\endgroup$ Dec 3, 2016 at 11:41
  • $\begingroup$ Thanks for the help. Managed to calculate the class of $T\mathbb{C}P^1$ due to your help. Still working on this bundle though, and I have one last question. In terms of real sections, the metric $g$ should be given as $(h+\overline{h})/2$. Which should give $g = (1+x^2+y^2) (dx^2 +dy^2)$. But then $\omega g + g \omega = dg$, so I get $\omega_{1}^1 = \omega_{2}^2 = d(1+x^2+y^2) / (1+x^2+y^2) = d(\log 1+x^2+y^2)$. Then $\Omega = d \omega + \omega \wedge \omega = 0$... So I'm finding the curvature form to be $0$. Any tips? $\endgroup$ Dec 5, 2016 at 8:57
  • $\begingroup$ Offhand, your strategy looks all right. I'd focus first on the $\omega \wedge \omega$ term, since wedging matrix-valued forms is a place where terms can easily appear to cancel when they don't. If that doesn't help, maybe add your calculation to your question? $\endgroup$ Dec 5, 2016 at 10:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.