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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 Agbb of Ag stabilizes as g 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 1 is an extension of (2r 1) by (0); in particular, it is not pure.

Keywords
Satake compactification, stable cohomology, mixed Hodge structure
Mathematical Subject Classification 2010
Primary: 14G35, 32S35
References
Publication
Received: 3 November 2015
Accepted: 10 August 2016
Published: 19 May 2017
Proposed: Benson Farb
Seconded: Jim Bryan, Dan Abramovich
Authors
Jiaming Chen
Institut de mathématiques de Jussieu-Paris Rive Gauche
Université Paris 7 Diderot
Campus de Grands Moulins
Bureau 745
Bâtiment Sophie Germain
75205 Paris Cedex 13
France
Eduard Looijenga
Yau Mathematical Sciences Center
Tsinghua University
Jin Chun Yuan West Building
Haidian District
Beijing, 100084
China