The stable cohomology of the Satake compactification of $\mathcal{A}_g$

Abstract

Charney and Lee have shown that the rational cohomology of the Satake–Baily–Borel compactification ${\mathcal{A}}_g^{bb}$ of ${\mathcal{A}}_g$ stabilizes as $g\to \infty$ and they computed this stable cohomology as a Hopf algebra. We give a relatively simple algebrogeometric proof of their theorem and show that this stable cohomology comes with a mixed Hodge structure of which we determine the Hodge numbers. We find that the mixed Hodge structure on the primitive cohomology in degrees $4r+2$ with $r\ge 1$ is an extension of $\mathbb{Q}\left(-2r-1\right)$ by $\mathbb{Q}\left(0\right)$; in particular, it is not pure.