interior of a triangle in $CP^2$

Question

In $\mathbb{C}\mathbb{P}^2$ we define coordinate triangle to be the one with sides $\{x_0=0\}, \{x_1=0\}$ and $\{x_2=0\}$. How would you define its interior? What kind of equation should it satisfy?

Answer 1

“Being interior” corresponds to some form of “ordering comparison”. Most usually you have some equation which is zero at the boundary, and you define that the interior is where the equation is positive. But you don't have a sign or an ordering for complex numbers. Therefore you cannot reasonably define interior.

You can use some function to turn complex numbers into real ones, e.g. by simply taking the real part only. But that will result in points being exterior and other points being interior with no boundary point in between these two. Except if you define boundary using this other function as well, in which case you'll no longer have the triangle boundary equations you stated in your question.