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>



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> I would like to know if its geometry has been studied in more detail in the sense of complex geometry (algebraic, differentiable, analytic). 
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References are appreciated since the question above mentions only Jantzen's "Lie Theory".



Has the geometry of the variety of nilpotent matrices over $\mathbb{C}$ been studied?
Consider $PGL\_n$ acting by conjugation on the space of $n\times n$ matrices $M\_n$, and let the GIT quotient map be $\pi:M\_n\to M\_n//PGL\_n$. I think you are asking about the geometry of the fibre of zero.


Regardless, I believe this is a special case of the nullcone for a reductive group action on a vector space $V$.


Two papers that come to mind about the geometry of the nullcone are:


1. *A Stratification of the Null Cone Via the Moment Map* by Linda Ness (appendix by David Mumford), American Journal of Mathematics, Vol. 106, No. 6 (Dec., 1984), pp. 1281-1329
2. *Irreducible components of the nullcone* by Richardson, R. W., Invariant theory, 409–434, Contemp. Math., 88, Amer. Math. Soc., Providence, RI, 1989.


The work of these two authors is worth reading if you are interested in these kinds of problems (and the references therein).


