The moduli space of $A_g$ of abelian varieties has three classical toroidal compactifications: (1) perfect, (2) 2nd Voronoi, and (3) Igusa blowup, each with its own distinct geometric meaning. It is an interesting problem to understand exactly how these compactifications are related. I will show that (1) and (2) are isomorphic in a neighborhood of the image of a regular map from the Deligne-Mumford's moduli space $\bar M_g$, and that the rational map $\bar M_g \to \bar A_g$ for (3) is not regular for $g>8$. This is a joint work with Adrian Brunyate.