Is there a name for the complex manifold given by $z_i\overline{z_i}=1$ ? If indeed it is a complex manifold.
I know it is homeomorphic to the real manifold is $S_{2n-1}$.
Also, is it true to say that:
"The isometries are O(2n) and the 'holomorphic isometries' are SU(n)?"
Is that the correct language to use? (If it is true)