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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?

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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 $(z,v)\mapsto z^{l_\infty}v$ –  domenico fiorenza Dec 12 '11 at 19:48
It's possible iff $L \cong {\mathcal O}(l_\infty - l_0)$. –  Allen Knutson Dec 15 '11 at 15:19
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