Parsing the Fibers of Critical Values

[WWGD]

Hi, All:

I am trying to better understand what happens at fibers of critical values:

Specifically, I had the fibers of the map:

f(z1,z2): ℂ²→ℂ , given by:

f(Z₁, Z₂)= Z₁²+z₂² =0

I rewrote it as a map from ℝ⁴→ℝ²=

f(x₁,y₁,x₂,x₂)=

(x₁²-y₁²+x₂²-

y₂² , x₁y₁+2x₂y₂)Which has the Jacobian:
(Sorry, I don't know how to Tex a matrix ):

[ 2x₁, 2y₁, 2x₂, 2y₂ ]

[ 2y₁, 2x₁, 2y₂, 2x₂


Now, by one of the family of theorems in { inverse function, implicit function, regular rank theorem} , the fibers of the non-zero values are submanifolds of ℂ².

After row-reducing, we get that the only critical value is (0,0) .

Now, some work on f^{-1}(0,0):

i)Not very helpful: Z₁²+Z₂²=0

implies Z₁=iZ₂.

ii) A little better:

Restricting to (x₁,x₂, 0, 0):

Then the general fiber is a circle, and the collection of all fibers is a cone .

Is there anything else we can do to get a better understanding of what the general

fibers are like ? I think all the regular fibers are isomorphic to each other--tho I cannot

find a good proof -- and then we have the critical fiber.

Any ideas?

Thanks.