#### Volume 21, issue 4 (2017)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
The stable cohomology of the Satake compactification of $\mathcal{A}_g$

### Jiaming Chen and Eduard Looijenga

Geometry & Topology 21 (2017) 2231–2241
##### Abstract

Charney and Lee have shown that the rational cohomology of the Satake–Baily–Borel compactification ${\mathsc{A}}_{g}^{bb}$ of ${\mathsc{A}}_{g}$ 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 $ℚ\left(-2r-1\right)$ by $ℚ\left(0\right)$; in particular, it is not pure.

##### Keywords
Satake compactification, stable cohomology, mixed Hodge structure
##### Mathematical Subject Classification 2010
Primary: 14G35, 32S35