I've been working on symplectic geometry, and I've encountered this two complex sets (manifolds) in $\mathbb{C}^4$:
$A_1 =\{ (z_1,z_2,z_3,z_4) \enspace | \enspace |z_1|^2+|z_3|^2=a , \enspace |z_2|^2 + |z_4|^2 = b \}$
$A_2 =\{ (z_1,z_2,z_3,z_4) \enspace | \enspace |z_1|^2+|z_3|^2=a , \enspace |z_1|^2 + |z_2|^2 + |z_4|^2 = b \}$
In both cases I need to find the manifold:
$$ M_i= A_i / T^2$$
where $T^2$ is the torus. I've read that these manifolds $M_i$ are "Hirzebruch" surfaces. No idea about it though. As I have never taken a course of complex surfaces or algebraic surfaces, sm question is what manifolds are $A_1$ and $A_2$? In order to compute the quotient. Or how can I show $M_i$ are Hirzebruch surfaces?
By the way the torus action considered is the standard hamiltonian action :
$$ (e^{i\theta_1},e^{i\theta_2},e^{i\theta_3},e^{i\theta_4}) \cdot (z_1,z_2,z_3,z_4)= (e^{i\theta_1} z_1,e^{i\theta_2} z_2,e^{i\theta_3} z_3,e^{i\theta_4} z_4).$$