This is extension of my previous question Moduli space about $CP^{N-1}$ and $T^* CP^{N-1}$., meaning of $\mathcal O(-1)$ in algebraic geometry? .

What i have been considered are followings First briefly state the problem, i want to find the moduli space of \begin{align} |\phi_1|^2 + |\phi_2|^2 - |\phi_3|^2 - |\phi_4|^2 =r \end{align} where $\phi$ is complex and the moduli space is the solutions of above equation with quotient of $U(1)$. (I already know the answer for this moduli space $\mathcal O(-1) \oplus \mathcal O(-1)$ over $CP^1$.

First checking its dimension, each $\phi$ has 2 real dimension, and constriant(above equation) and quotient of $U(1)$ gives each 1 degrees of freedom, thus \begin{align} 8-1-1=6 \end{align} which gives 6 real dimension, $i.e$ 3 complex dimension. For $\mathcal O(-1) \oplus \mathcal O(-1)$ over $CP^1$. each taughtological line bundle has $1$ complex dimension and $CP^1$ has 1 complex dimension thus its space has consist of 3 complex dimension.

Nextly to construct, local trivialization, divide this space as $U_0$ and $U_1$. where $U_0 = \{ z_0 = \frac{\phi_4}{\phi_3} \in \mathbf{C}, \,\ \phi_3 \neq 0\}$. and $U_1 = \{ z_1 = \frac{\phi_3}{\phi_4} \in\mathbf{C}, \,\ \phi_4 \neq 0 \}$.

For $U_0$, from the equation \begin{align} &|\phi_1|^2 + |\phi_2|^2 -(1+ |z_0|^2)|\phi_3|^2 =r \\ & |\phi_1|^2 = r + (1+ |z_0|^2)|\phi_3|^2 -|\phi_2|^2 \end{align} where the $\phi_1$ is constrcuted by $\phi_2$, $\phi_3$ freely. Thus i see following local trivalization \begin{align} \Phi_{U_0} : E \rightarrow &U_0 \times C^2 \\ & \left( \frac{\phi_4}{\phi_3}, (\phi_2, \phi_3) \right)=\left( z_0, (\phi_2, \phi_3) \right) \end{align}

And do similar things on $U_1$ i see \begin{align} \Phi_{U_1} : E \rightarrow &U_1 \times C^2 \\ & \left( \frac{\phi_3}{\phi_4}, (\phi_2, \phi_4) \right) = \left( z_1, (\phi_2, \phi_4) \right) \end{align}

The next task which left to me is find the transition map between $\Phi_{U_0}$ and $\Phi_{U_1}$.

I know for $\mathcal O(k)$ over $CP^1$ cases; $i.e$, by taking local trivialization for $U_0 \times C$, and $U_1 \times C$. Then formulating inverse relation and their transition order, we can find its transition function. see shanyuji-Geometry lecture note 5,

Can you give me some explict transition function for the above case($U_i \times C^2$)? $i.e$ what i known from above, similar with reference set $z_0 = \frac{1}{z_1}$. which is exchange of $\phi_3 \leftrightarrow \phi_4$. but in this case still $\phi_2$ exists as a free variable. Thus i failed up constructing transition function betwen them....

Any comment or answer will be helpful in extending understanding of this topics. Thanks


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.