Consider the complex projective variety given by $X^n = 0$, where $X\in \mathrm{M}_n(\mathbb{C})$ and, say, $n\geq 3$. Some basic properties of it are already mentioned in this question:

https://math.stackexchange.com/questions/405291/variety-of-nilpotent-matrices

I would like to know if its geometry has been studied in more detail in the sense of complex geometry (algebraic, differentiable, analytic).

References are appreciated since the question above mentions only Jantzen's "Lie Theory".