First, 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.

Many references related to Mirror symmetry, cover this for granted. I want to understand the meaning precisely.

cf) relevant question already asked by me [Moduli space for $CP^N$ and $T^{*} CP^N$ in $\mathcal{N}=2$ SUSY ] , and [ meaning of $\mathcal O(-1)$ in algebraic-geometry]



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