The stable cohomology of the Satake compactification of \(\mathcal A_g\) (Q2356963)

Denote by \(\mathcal A_g\) the moduli space of principally polarized abelian varieties (considered as a complex analytic space), and let \(\mathcal A_g^{\mathrm{bb}}\) be its Satake-Baily-Borel compactification. \textit{R. Charney} and \textit{R. Lee} [Topology 22, 389--423 (1983; Zbl 0562.32017)]have shown that the rational cohomology \(H^k (\mathcal A_g^{\mathrm{bb}}, \mathbb{Q})\) stabilizes as \(g \to \infty\), and computed the stable cohomology \(H^{\bullet} (\mathcal A_{\infty}^{\mathrm{bb}}, \mathbb{Q})\) as a Hopf algebra. In this paper the authors show that \(H^{\bullet} (\mathcal A_{\infty}^{\mathrm{bb}}, \mathbb{Q})\) comes with a mixed Hodge structure, and determine the Hodge numbers. Along the way, they also give a new algebro-geometric proof of the theorem of \textit{R. Charney} and \textit{R. Lee} [Topology 22, 389--423 (1983; Zbl 0562.32017)].