Mathematische Annalen

, Volume 369, Issue 3–4, pp 1353–1381 | Cite as

The intersection cohomology of the Satake compactification of \({\mathcal {A}}_g\) for \(g \le 4\)



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.

Mathematics Subject Classification

Primary 14K10 Secondary 14F43 55N33 



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.


  1. 1.
    de Cataldo, M.A.: Perverse sheaves and the topology of algebraic varieties. Five lectures at the 2015 PCMI. arXiv:1506.03642
  2. 2.
    de Cataldo, M.A., Migliorini, L.: The hard Lefschetz theorem and the topology of semismall maps. Ann. Sci. École Norm. Sup. 35(5), 759–772 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    de Cataldo, M.A., Migliorini, L.: The Hodge theory of algebraic maps. Ann. Sci. École Norm. Sup. 38(5), 693–750 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    de Cataldo, M.A., Migliorini, L.: The decomposition theorem, perverse sheaves and the topology of algebraic maps. Bull. Am. Math. Soc. (N.S.) 46(4), 535–633 (2009)Google Scholar
  5. 5.
    Charney, R., Lee, R.: Cohomology of the Satake compactification. Topology 22(4), 389–423 (1983)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Erdenberger, C.: A finiteness result for Siegel modular threefolds. Dissertation, Hannover (2008).
  7. 7.
    Erdenberger, C., Hulek, K., Grushevsky, S.: Intersection theory of toroidal compactifications of \({\cal{A}}_4\). Bull. Lond. Math. Soc. 38(3), 396–400 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Esnault, H., Viehweg, E.: Chern classes of Gauss-Manin bundles of weight 1 vanish. K Theory 26(3), 287–305 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    van der Geer, G.: The Chow ring of the moduli space of abelian threefolds. J. Algebraic Geom. 7(4), 753–770 (1998)MathSciNetMATHGoogle Scholar
  10. 10.
    van der Geer, G.: Cycles on the moduli space of abelian varieties, pp. 65–89. Aspects Math., E33. Vieweg, Braunschweig (1999)Google Scholar
  11. 11.
    Goresky, M., Harder, G., MacPherson, R.: Weighted cohomology. Invent. Math. 116(1–3), 139–213 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Goresky, M., Harder, G., MacPherson, R., Nair, A.: Local intersection cohomology of Baily-Borel compactifications. Compos. Math. 134(2), 243–268 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Goresky, M., MacPherson, R.: Intersection homology II. Invent. Math. 72(1), 77–129 (1983)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Goresky, M., Pardon, W.: Chern classes of automorphic vector bundles. Invent. Math. 147(3), 561–612 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Grushevsky, S., Hulek, K., Tommasi, O.: Stable cohomology of perfect cone compactifications of \({\cal{A}}\). J. Reine Angew. Math. (2016). doi: 10.1515/crelle-2015-0067
  16. 16.
    Hain, R.: The rational cohomology ring of the moduli space of abelian 3-folds. Math. Res. Lett. 9(4), 473–491 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hoffman, J.W., Weintraub, S.H.: The Siegel modular variety of degree two and level three. Trans. Am. Math. Soc. 353(8), 3267–3305 (2001)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hulek, K., Sankaran, G.K.: The nef cone of toroidal compactifications of \({\cal{A}}_4\). Proc. Lond. Math. Soc. 88(3), 659–704 (2004)Google Scholar
  19. 19.
    Hulek, K., Tommasi, O.: Cohomology of the toroidal compactification of \({\cal{A}}_3\). Vector bundles and complex geometry, Contemp. Math, vol. 522, pp. 89–103. American Mathematical Society, Providence, RI (2010)Google Scholar
  20. 20.
    Hulek, K., Tommasi, O.: Cohomology of the second Voronoi compactification of \({\cal{A}}_4\). Doc. Math. 17, 195–244 (2012)MathSciNetMATHGoogle Scholar
  21. 21.
    Kirwan, F.: Rational intersection cohomology of quotient varieties. Invent. Math. 86(3), 471–505 (1986)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Looijenga, E.: \(L^2\)-cohomology of locally symmetric varieties. Compos. Math. 67(1), 3–20 (1988)MathSciNetMATHGoogle Scholar
  23. 23.
    Looijenga, E.: Goresky-Pardon extensions of Chern classes and associated Tate extensions. arXiv:1510.04103 [math.AG]
  24. 24.
    Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and geometry, vol. II. Progr. Math., vol. 36, pp. 271–328. Birkhäuser Boston, Boston, MA (1983)Google Scholar
  25. 25.
    Saper, L., Stern, M.: \(L_2\)-cohomology of arithmetic varieties. Ann. Math. 132(1), 1–69 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematics DepartmentStony Brook UniversityStony BrookUSA
  2. 2.Institut für Algebraische GeometrieLeibniz Universität HannoverHannoverGermany

Personalised recommendations