Visualizing $f: \mathbb{C}^2\to \mathbb{C}, (z_1,z_2) \mapsto z_1^2+z_2^2$ and its Hypersurface

Question

I have difficulties to visualize the graph of the given function $$f:\mathbb{C}^2\to \mathbb{C}, (z_1,z_2) \mapsto z_1^2+z_2^2$$ and the Hypersurface $$ X_0 = \{(z_1,z_2) \in \mathbb{C}^2 \mid z_1^2+z_2^2 = 0\}$$

Apparently the Hypersurface looks like two crossed complex lines. But unfortunately I don't understand how to get to its picture. I have to admit, that I am not too comfortable with complex analysis but i'd really love to catch up.

Could someone help me visualizing it and maybe even show me how to draw the Hypersurface's picture?

I'd really appreciate any help. Thank you very much.

Answer 1

We can write the defining polynomial as $(z_1 + iz_2)(z_1 - iz_2) = z_1^2 + z_2^2$, such that you get $X_0 = V(z_1 + iz_2) \cup V(z_1 - iz_2).$

Can you see lines now?

PS.: In case that the notion is not familiar to you, $V(f) = \lbrace (z_1,\dotsc,z_n) \in \mathbb{C}^n \mid f(z_1,\dotsc,z_n) = 0 \rbrace$ is just the zero/vanishing set of some polynomial $f \in \mathbb{C}[T_1,\dotsc,T_n]$.