The goal of this paper is to generalize the results of [L. Clozel and E. Ullmo, Ann. Math. (2) 161, No. 3, 1571–1588 (2005; Zbl 1099.11031)] concerning the equi-distribution of special subvarieties of Shimura varieties.
Given a connected Shimura variety \(S\), one has the notion of special points, and of special subvarieties. Roughly speaking, a special subvariety is a sub-Shimura variety. Every special subvariety \(Z\subseteq S\) carries a probability measure \(\mu_Z\) with support \(Z\).
Whenever \(Z\) is not of the form
\[ Z = M_1 \times \{ x \} \subseteq M = M_1 \times M_2 \subseteq S \]
for a special subvariety \(M\), then it is called “NF” (non facteur).
The author shows the following theorem: Let \((M_n)_n\) be a sequence of NF special subvarieties of \(S\). Then there exists an NF special subvariety \(Z\subseteq S\) and a subsequence \((\mu_{M_n})_{n\in\mathbb N}\), \(N\subseteq \mathbb N\), of the sequence of corresponding measures which converges weakly to \(\mu_Z\). Furthermore, for \(n\in\mathbb N\) sufficiently large, \(M_n\subseteq Z\).
It is clear that in the theorem one cannot replace “NF special” by “special”. For instance, the special points (i.e. the \(0\)-dimensional special subvarieties) are dense in \(S\) for the complex topology.
From the theorem, one can deduce the following result: Let \(Y\) be a subvariety of \(S\). There exists a finite set \(M_i\) of NF special subvarieties of \(S\), such that \(M_i\subseteq Y\) for all \(i\), and such that every NF special subvariety contained in \(Y\) is in fact contained in the union \(\bigcup M_i\). This result should be seen in the light of the André-Oort conjecture which states that the same should be true with “NF special” replaced by “special” everywhere. Compare also the work of Edixhoven and Yafaev in this direction.
As for loc. cit., the proofs of the paper at hand rely on results in the ergodic theory of unipotent flows by Ratner and by Mozes and Shah.