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Is this quotient of $\mathbb{C}^{m+1}$ by $U(1)$ only "nice" for $m=1$?
Let $V^{m+1} = \mathbb{C}^{m+1}$ and let $U(1)$ act on it by its diagonal representation, so that really, it is just like scalar multiplication by a unit modulus complex number.
I am interested in the ...
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