# Ampleness of the Canonical Bundle for Siegel Modular Varieties

Background

Throughout I only work with varieties over $\mathbb{C}$.

For $p$ a prime number, Let $Y(p)$ denote the modular curve parametrizing elliptic curves together with full $p$-torsion structure, and $X(p)$ denote its smooth compactification. Then $X(p)$ has a boundary divisor $D$ given by the cusps $X(p)\backslash Y(p)$.

Now, it is easy to see by computing genus that for sufficiently large $p$, the canonical class $K$ on $X(p)$ is ample.

Moreover, for any fixed integer $N>0$, for sufficiently large $p$ the divisor $K-N\cdot D$ is ample on $X(p)$.

My Question

Basically, is the above true in higher dimension? Specifically, consider $A_g(p)$ to be the moduli space of principally polarized abelian varieties together with a full $p$-torsion structure, and $\overline{A_g(p)}$ a smooth toroidal compactification. Then is the canonical bundle of $\overline{A_g(p)}$ ample for large enough $p$? Better, can you show that $K-N\cdot D$ is ample for any $N$ and large enough $p$?

Some relevant facts I happen to know

I know that Mumford, Tai, Hizerbruch etc. showed that $K$ is big for large enough $p$ (i.e. $\textrm{dim } H^0(nK)$ grows like a polynomial in $n$ of the right degree.)

The way they do this is to compute sections of $K$ on the open part using Hizerbruch proportionality (+ Mumfords extension of it to non-compact settings) and then just compute fourier expansions at the cusps to show that many of these sections extend. This method seems very elegant and clear to me, but it seems harder to actually get ampleness.

• To my knowledge, the answer is known only for $g=2$ and $3$, see Hulek-Sankaran, The Geometry of Siegel Modular Varieties, arXiv:9810153. I am not aware of any progress after that paper (but I am not an expert). – abx Apr 5 '14 at 14:31