Complex Manifold question

Question

So let's take the following:

$$C\subset \mathbb{P}^2(\mathbb{C})$$ where $$C = \left\{[x_0:x_1:x_2]; x_0^2 + x_1^2 + x_2^2 = 0 \right\}$$

How can I proceed to prove that $C$ is a complex manifold? ($\mathbb{P}$ denotes the Projective Space).

Answer 1

I thought I could have answered my own question, instead of deleting it, for it can help someone in the future. Hope it's all right.

In $\mathbb{P}^2(\mathbb{C})$ we want $Z(x_0^2 + x_1^2 + x_2^2)$ to be holomorphic. ($Z(\cdot)$ denotes the place of the zeroes of $\cdot$).

We then have $Z(1 + x^2 + y^2)$ where $$x = \dfrac{x_1}{x_0} ~~~~~~~ y = \dfrac{x_2}{x_0}$$

The Jacobian $J(1+x^2+y^2)$ has the maximum rank out from $(0, 0)$ but $(0, 0) \notin Z(1+x^2 +y^2)$.

Whence $C$ is a complex manifold according to the implicit holomorphic function theorem.