# Minimal compactification

In their famous book, Faltings and Chai constructed, among other things, minimal and toroidal compactification of the Siegel moduli space. The reason for the use of the word toroidal is clear, since toroidal geometry is used to define such compactification, it is not clear to me the use of the word minimal. So the question is:

in which sense the minimal compactification is minimal?

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Given the symmetric space $D$ of non-compact type for the real semi-simple Lie group $G$, there are $2^r-1$ Satake compactifications $\bar{D}$ for $D$ up to homeomorphism, where $r$ is the real rank of $G$.

These compactifications correspond to the non-empty subsets of a set $S$ of simple roots, and as such they form a semi-lattice: if $S_1 \subset S_2$, then the identity of $D$ extends to a continuous mapping $\bar{D}_{S_2} \to \bar{D}_{S_1}$.

The Satake-Baily–Borel compactification is one of the minimal (in the semi-lattice sense) Satake compactifications in the case where $D$ is Hermitian, i.e., has a $G$-invariant complex structure, see this entry in the Encyclopedia of Mathematics.

According to [Faltings-Chai, Degeneration of Abelian Varieties, p. 136], the construction of the minimal compactification of $\mathcal{A}_g$ mimics the construction of the the Satake-Baily-Borel compactification of a symmetric space, and this is why the word "minimal" is used.

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I see. Thank you! – user32948 Apr 9 '13 at 13:00
Actually $A_g$ is a hermitian symmetric space, so the minimal compactification doesn't just mimic the construction of Satake-Baily-Borel: it is a special case of it. – Dan Petersen Apr 10 '13 at 18:37

There is another reason. If $X$ is a Shimura variety (eg the Siegel modular variety) and $X^*$ is its minimal (or Baily-Borel, or Baily-Borel-Satake) compactification, then it has the following property : for every other open embedding with dense image $X\rightarrow \overline{X}$ with $\overline{X}-X$ a divisor with normal crossings, there exists a unique map (of algebraic varieties) $\overline{X}\rightarrow X^*$ compatible with the embeddings of $X$ in $X^*$ and $\overline{X}$. Cf the remark after corollary 3.16 of Milne's "Introduction to Shimura varieties" (http://jmilne.org/math/xnotes/svi.pdf).

(Note that $X^*-X$ itself is not a divisor with normal crossings, except in very particular cases like modular curves.)

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