If $F$ is a totally real number field with $[F:\mathbb{Q}] = d>1$, $X$ is the moduli space of Hilbert-Blumenthal Abelian varieties for $F$, and $\overline{X}$ is the projective toroidal compactification, then a paper I am reading refers to $D = \overline{X} \setminus X$ as the boundary divisor. But the cusps of $X$ should be points, so $D$ should be a collection of points, which are too high codimension to be called a divisor. So what does this mean?


I have the suspicion that the "projective toroidal compactification" might be something other than what I expect (i.e. not just the union of $X$ with the cusps), so that the boundary will actually be of codimension 1. But it may also be that the boundary divisor refers to the sum of the codimension $1$ pieces, in which case it could just be trivial if all of the irreducible components are of high codimension.



Boundary divisor of projective toroidal compactification
Toroidal compactifications of Shimura varieties are smooth, and $D$ will certainly be of codimension one (when it is not empty). There are in fact a number of different compactifications of Shimura varieties to consider each with different desirable properties. For example, there is a smaller minimal compactification which has the property that the sheaf $\omega$ suitably defined extends to an ample bundle, which is nice, but this is usually a singular space, which is bad, but it at least a projective variety, which is good, but the frogurt is cursed, which is bad. If you "complete" your spaces in too brutal a manner it's not clear you even get projective varieties. These compactifications basically only coincide in the simplest cases like the modular curve.


A useful reference to sort this out might be as follows:


<https://www.math.ias.edu/~goresky/pdf/compactifications.pdf>


