# On toroidal compactifications of Hilbert Kuga-Sato varieties

Let $F$ be a totally real field of degree d. There are Hilbert modular varieties over $\mathbb{Q}$ that paramatrize abelian varieties of dimension d with an action of $\mathcal{O}_F$ the ring of integers of $F$ and some level structure.

Suppose we fix some level and we consider $\mathcal{A}\to M$, the universal abelian variety and for any positive integer $n$ the $n$-fold fiber product $\mathcal{A}^n\to M$ over $M$. This Kuga-Sato variety comes endowed with an action of $M_{n\times n}(\mathcal{O}_F)$. Do you know if these endomorphisms extend to the smooth toroidal compactifications denoted by $\overline{\mathcal{A}^n}$ in http://math.univ-lille1.fr/~mladen/articles/Pad16-DiTi.pdf ?

• Naive question: do you know the answer in the case $n=1$? Jun 16, 2017 at 19:57
• Unfortunatly no. In fact, if I knew it holds for $n=1$, I could probably get an action of $(\mathcal{O}_F^\times)^n\rtimes S_n$ on some desingularization of $\left(\overline{\mathcal{A}}\right)^n$ using reasolution of singularities. Anyway, I know that what I want is true $\forall n$ in the case when $F=\mathbb{Q}$ because Deligne and Scholl constructed canonical desingularizations for $\left(\overline{\mathcal{E}}\right)^n$ for $\overline{\mathcal{E}}$ the universal generalized elliptic curve.
• Actually one could only get an action of $\{\pm1\}^n\rtimes S_n$ using resolution of singularities because $(\mathcal{O}_F^\times)^n\rtimes S_n$ is not a finite group.