**First, i want to know what is $\mathcal O(-1)$ bundle is. (Definition or geometric interpretation, and so on)**

**Second** I want understand how the following is constructed.

Consider a $\phi$ as a coordinates on a copy of $Z= C^N$

Then, I know
\begin{align}
|\phi_1|^2 + |\phi_2|^2 + \cdots |\phi_N|^2 = r
\end{align}
which describe $S^{2N-1}$.

Implementing $U(1)$ condition the space of solution is described by

\begin{align}
CP^N = S^{2N+1}/U(1)
\end{align}
Now consider slightly different case
\begin{align}
|\phi_1|^2 + |\phi_2|^2 - |\phi_3|^2 - |\phi_4|^2 =r
\end{align}
this gives **$\mathcal O(-1) \oplus \mathcal O(-1)$ over $CP^1$.
I want to know what this means and how to obtain.**

This post imported from StackExchange Mathematics at 2015-10-31 22:13 (UTC), posted by SE-user phy_math