Abstract
We completely determine the intersection cohomology of the Satake compactifications \({{\mathcal {A}}_{2}^{\mathrm{Sat}}},{{\mathcal {A}}_{3}^{\mathrm{Sat}}}\), and \({{\mathcal {A}}_{4}^{\mathrm{Sat}}}\), except for \({ IH}^{10}({{\mathcal {A}}_{4}^{\mathrm{Sat}}})\). We also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the Satake compactification in these genera. As a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved.
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Acknowledgments
We would like to thank Mark Goresky and Luca Migliorini very much for generously explaining to us at the Institute for Advanced Study the many details about intersection cohomology and the decomposition theorem. We are grateful to Eduard Looijenga for numerous enlightening discussions, in particular about the extension of tautological classes to the Satake compactification. We are especially indebted to Mark Goresky for providing the proof of Proposition 5.1 and to Luca Migliorini for detailed comments on a preliminary version of this manuscript. We thank the referee for a careful reading of the manuscript and suggested improvements of the exposition. Both authors thank the Institute for Advanced Study and the Fund for Mathematics for support and the excellent working conditions in Spring 2015, when this work was begun. The first author is grateful to the Alexander von Humboldt foundation for its support; this work was partially enabled by the Friedrich Wilhelm Bessel Research Award from the Humboldt foundation.
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Research of S. Grushevsky is supported in part by National Science Foundation under the Grants DMS-12-01369 and DMS-15-01265, and by a Simons Fellowship in Mathematics (Simons Foundation Grant #341858 to Samuel Grushevsky). Research of K. Hulek is supported in part by DFG Grant Hu-337/6-2.
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Grushevsky, S., Hulek, K. The intersection cohomology of the Satake compactification of \({\mathcal {A}}_g\) for \(g \le 4\) . Math. Ann. 369, 1353–1381 (2017). https://doi.org/10.1007/s00208-016-1491-1
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DOI: https://doi.org/10.1007/s00208-016-1491-1