0
$\begingroup$

Let $\mathbb{P}(V) = \mathbb{P}(\mathbb C \oplus \mathbb C)$ be with a $\mathbb C^*$ action : $\lambda (u,v) = (u,\lambda v)$. There are two fixed points of this action, say $0$ and $\infty$. What does this mean to equivariantly lift this action to a line bundle $L$ over $\mathbb P(V)$, by choosing the weights $[l_0, l_{\infty}]$ of the fibre representations $L_0, L_{\infty}$ at the fixed points?

$\endgroup$
  • $\begingroup$ it just means that on the fiber at $0$ the $\mathbb{C}^*$-action is $(z,v)\mapsto z^{l_0}v$ and on the fiber at $\infty$ the $\mathbb{C}^*$-action is $(z,v)\mapsto z^{l_\infty}v$ $\endgroup$ – domenico fiorenza Dec 12 '11 at 19:48
  • $\begingroup$ It's possible iff $L \cong {\mathcal O}(l_\infty - l_0)$. $\endgroup$ – Allen Knutson Dec 15 '11 at 15:19

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.