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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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Volume 2018, Issue 741

# Stable cohomology of the perfect cone toroidal compactification of 𝒜g

Samuel Grushevsky
/ Klaus Hulek
/ Orsola Tommasi
Published Online: 2015-12-17 | DOI: https://doi.org/10.1515/crelle-2015-0067

## Abstract

We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification ${\mathcal{𝒜}}_{g}^{\mathrm{Perf}}$ of the moduli space of complex principally polarized abelian varieties stabilizes in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification ${\mathcal{𝒜}}_{g}^{\mathrm{Matr}}$ stabilizes in fixed degree, and forms a polynomial algebra. For degree up to 8, we describe explicitly the generators of the cohomology, and discuss various approaches to computing all of the stable cohomology in general.

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Revised: 2015-06-27

Published Online: 2015-12-17

Published in Print: 2018-08-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-12-01369

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: Hu-337/6-1

Award identifier / Grant number: Hu-337/6-2

Research of the first author is supported in part by National Science Foundation under the grant DMS-12-01369. Research of the second and third authors is supported in part by DFG grants Hu-337/6-1 and Hu-337/6-2. The final revision of the paper was completed at the Institute for Advanced Study at Princeton where the second author was supported by the Fund for Mathematics.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 741, Pages 211–254, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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