When you deal with statified pseudomanifolds, you also have a concept of normal pseudomanifolds. A pseudomanifold $X$ of dimension $n$ is said to be normal if for every point $x\in X$ the local homology group $H_n(X,X-x,\mathbb{Z})$ is isomorphic to $\mathbb{Z}$. Now you know that any complex algebraic variety has a structure of stratified pseudomanifold. Using Zariski’s Main Theorem, one can prove that a normal complex algebraic variety is a normal pseudomanifold.
If you consider a triangulation $T$ of $X$ ($dim(X)=2$) then you can also prove that $X$ is normal if and only if the link of eack simplex in the $n-2$-skeleton of $T$ is connected.
This is proved in Goresky, MacPherson "Intersection Homology theory" (Topology Vol. 19 (1980)).