I am trying to understand an example that my professor introduced. At the time, I thought that I understood it:
Consider $\mathbb{CP}^2$ with homogeneous coordinates $X_0$, $X_1$, and $X_2$ and let $C\subset\mathbb{CP}^2$ satisfy $X_0X_1+X_0X_2+X_1X_2=0$.
Is $C$ an analytic subvariety?
We defined an analytic subvariety of a complex manifold as "a subset given locally as the zeros of a finite collection of holomorphic functions".
Since $P(X_0,X_1,X_2)=X_0X_1+X_0X_2+X_1X_2$ is homogeneous of degree $2$, we have that $C$ is well-defined.
I believe that it suffices to show that $P$ is holomorphic, but I am having trouble:
Let $(U_0,\varphi_0)$ be a standard chart. Then $\left(P\circ\varphi_0^{-1}\right)\left(\left(X_1,X_2\right)\right)=P\left(\left[1,X_1,X_2\right]\right)$ is not well-defined on $U_0$.
Am I missing something?
Here, $U_0=\{[X_0,X_1,X_2]:X_0\neq0\}$ and $\varphi_0:U_0\to\mathbb C^2$ is defined by
$$\left[X_0,X_1,X_2\right]\mapsto\left(\frac{X_1}{X_0},\frac{X_2}{X_0}\right).$$