Mathematische Annalen

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

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

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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.

Mathematics Subject Classification

Primary 14K10 Secondary 14F43 55N33 

References

  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). http://d-nb.info/98938540X/34
  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

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