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Journal fĂĽr die reine und angewandte Mathematik

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Volume 2018, Issue 741


Stable cohomology of the perfect cone toroidal compactification of đť’śg

Samuel Grushevsky / Klaus Hulek
  • Institut fĂĽr Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30060 Hannover, Germany
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/ Orsola Tommasi
Published Online: 2015-12-17 | DOI: https://doi.org/10.1515/crelle-2015-0067


We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification đť’śgPerf 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 đť’śgMatr 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.


  • [1]

    V. Alexeev and A. Brunyate, Extending Torelli map to toroidal compactifications of Siegel space, Invent. Math. 188 (2011), no. 1, 175–196. Web of ScienceGoogle Scholar

  • [2]

    A. Ash, D. Mumford, M. Rapoport and Y. Tai, Smooth compactification of locally symmetric varieties, 2nd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge 2010. Google Scholar

  • [3]

    E. Baranovskii and V. Grishukhin, Non-rigidity degree of a lattice and rigid lattices, European J. Combin. 22 (2001), no. 7, 921–935. CrossrefGoogle Scholar

  • [4]

    G. Barthel, J.-P. Brasselet, K.-H. Fieseler, O. Gabber and L. Kaup, Relèvement de cycles algébriques et homomorphismes associés en homologie d’intersection, Ann. of Math. (2) 141 (1995), no. 1, 147–179. CrossrefGoogle Scholar

  • [5]

    A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 235–272. CrossrefGoogle Scholar

  • [6]

    A. Borel, Stable real cohomology of arithmetic groups II, Manifolds and groups. Papers in honor of Yozo Matsushima, Progr. Math. 14, Birkhäuser, Basel (1981), 21–55. Google Scholar

  • [7]

    R. Charney and R. Lee, Cohomology of the Satake compactification, Topology 22 (1983), no. 4, 389–423. Web of ScienceCrossrefGoogle Scholar

  • [8]

    J. Chen and E. Looijenga, On the stable cohomology of the Satake compactification and its mixed Hodge structure, in preparation.

  • [9]

    D. Cox, J. Little and H. Schenck, Toric varieties, Grad. Stud. Math. 124, American Mathematical Society, Providence 2011. Google Scholar

  • [10]

    P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. Inst. Hautes Études Sci. 35 (1968), 259–278. Google Scholar

  • [11]

    C. Deninger and J. Murre, Motivic decomposition of abelian schemes and the Fourier transform, J. reine angew. Math. 422 (1991), 201–219. Google Scholar

  • [12]

    M. Deza and V. Grishukhin, Nonrigidity degree of root lattices and their duals, Geom. Dedicata 104 (2004), 15–24. CrossrefGoogle Scholar

  • [13]

    A. Durfee, Intersection homology Betti numbers, Proc. Amer. Math. Soc. 123 (1995), no. 4, 989–993. CrossrefGoogle Scholar

  • [14]

    M. Dutour Sikirić, K. Hulek and A. Schürmann, Smoothness and singularities of the perfect form compactification of 𝒜g, Algebraic Geom. 2 (2015), no. 5, 642–653. Google Scholar

  • [15]

    D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Grad. Texts in Math. 150, Springer, New York 1995. Google Scholar

  • [16]

    P. Elbaz-Vincent, H. Gangl and C. Soulé, Perfect forms, K-theory and the cohomology of modular groups, Adv. Math. 245 (2013), 587–624. CrossrefWeb of ScienceGoogle Scholar

  • [17]

    R. Erdahl and S. Ryshkov, The empty sphere. II, Canad. J. Math. 40 (1988), no. 5, 1058–1073. CrossrefGoogle Scholar

  • [18]

    R. Erdahl and S. Ryshkov, On lattice dicing, European J. Combin. 15 (1994), no. 5, 459–481. CrossrefGoogle Scholar

  • [19]

    G. Faltings and C.-L. Chai, Degeneration of abelian varieties, Ergeb. Math. Grenzgeb. (3) 22, Springer, Berlin 1990. Google Scholar

  • [20]

    W. Fulton, Intersection theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin 1998. Google Scholar

  • [21]

    W. Fulton and J. Harris, Representation theory, Grad. Texts in Math. 129, Springer, New York 1991. Google Scholar

  • [22]

    G. van der Geer, The Chow ring of the moduli space of abelian threefolds, J. Algebraic Geom. 7 (1998), no. 4, 753–770. Web of ScienceGoogle Scholar

  • [23]

    J. Giansiracusa and G. Sankaran, personal communication, 2013.

  • [24]

    S. Grushevsky, Geometry of 𝒜g and its compactifications, Algebraic geometry—Seattle 2005. Part 1, Proc. Sympos. Pure Math. 80, American Mathematical Society, Providence (2009), 193–234. Google Scholar

  • [25]

    S. Grushevsky and K. Hulek, Principally polarized semiabelic varieties of torus rank up to 3, and the Andreotti–Mayer loci, Pure Appl. Math. Q. 7 (2011), 1309–1360. CrossrefGoogle Scholar

  • [26]

    S. Grushevsky and K. Hulek, The class of the locus of intermediate Jacobians of cubic threefolds, Invent. Math. 190 (2012), no. 1, 119–168. Web of ScienceCrossrefGoogle Scholar

  • [27]

    S. Grushevsky and K. Hulek, Geometry of theta divisors—A survey, A celebration of algebraic geometry, Clay Math. Proc. 18, American Mathematical Society, Providence (2013), 361–390. Google Scholar

  • [28]

    R. Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997), no. 3, 597–651. CrossrefGoogle Scholar

  • [29]

    R. Hain, The rational cohomology ring of the moduli space of abelian 3-folds, Math. Res. Lett. 9 (2002), no. 4, 473–491. CrossrefGoogle Scholar

  • [30]

    R. Hain, Normal functions and the geometry of moduli spaces of curves, Handbook of moduli. Vol. I, International Press, Somerville (2013), 527–578. Google Scholar

  • [31]

    J. L. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. (2) 121 (1985), no. 2, 215–249. CrossrefGoogle Scholar

  • [32]

    A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge 2002. Google Scholar

  • [33]

    K. Hulek and O. Tommasi, Cohomology of the toroidal compactification of 𝒜3, Vector bundles and complex geometry, Contemp. Math. 522, American Mathematical Society, Providence (2010), 89–103. Google Scholar

  • [34]

    K. Hulek and O. Tommasi, Cohomology of the second Voronoi compactification of 𝒜4, Doc. Math. 17 (2012), 195–244. Google Scholar

  • [35]

    E. Looijenga and V. Lunts, A Lie algebra attached to a projective variety, Invent. Math. 129 (1997), no. 2, 361–412. CrossrefGoogle Scholar

  • [36]

    I. Madsen and M. Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture, Ann. of Math. (2) 165 (2007), no. 3, 843–941. Web of ScienceCrossrefGoogle Scholar

  • [37]

    M. Melo and F. Viviani, Comparing perfect and 2nd Voronoi decompositions: The matroidal locus, Math. Ann. 354 (2012), no. 4, 1521–1554. CrossrefWeb of ScienceGoogle Scholar

  • [38]

    B. Moonen, On the Chow motive of an abelian scheme with non-trivial endomorphisms, J. reine angew. Math. (2014), 10.1515/crelle-2013-0115. Web of ScienceGoogle Scholar

  • [39]

    D. Mumford, Hirzebruch’s proportionality theorem in the noncompact case, Invent. Math. 42 (1977), 239–272. CrossrefGoogle Scholar

  • [40]

    D. Mumford, On the Kodaira dimension of the Siegel modular variety, Algebraic geometry—Open problems (Ravello 1982), Lecture Notes in Math. 997, Springer, Berlin (1983), 348–375. Google Scholar

  • [41]

    Y. Namikawa, Toroidal compactification of Siegel spaces, Lecture Notes in Math. 812, Springer, Berlin 1980. Google Scholar

  • [42]

    C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge structures, Ergeb. Math. Grenzgeb. (3) 52, Springer, Berlin 2008. Google Scholar

  • [43]

    N. Shepherd-Barron, Perfect forms and the moduli space of abelian varieties, Invent. Math. 163 (2006), no. 1, 25–45. CrossrefGoogle Scholar

  • [44]

    G. Thompson, Skew invariant theory of symplectic groups, pluri-Hodge groups and 3-manifold invariants, Int. Math. Res. Not. IMRN 2007 (2007), no. 15, Article ID rnm048. Google Scholar

  • [45]

    F. Vallentin, Sphere coverings, lattices, and tilings (in low dimensions), PhD thesis, Center for Mathematical Sciences, Munich University of Technology, Munich 2003. Google Scholar

  • [46]

    C. Voisin, Chow rings, decomposition of the diagonal and the topology of families, Ann. of Math. Stud. 187, Princeton University Press, Princeton 2014. Google Scholar

  • [47]

    G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire. Sur quelques propriétés des formes quadratiques positives parfaites, J. reine angew. Math. 134 (1908), 79–178. Google Scholar

  • [48]

    G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les paralléloèdres primitifs, J. reine angew. Math. 134 (1908), 198–287. Google Scholar

  • [49]

    G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Seconde partie: domaines de formes quadratiques correspondants aux différents types deparalléloèdres primitifs, J. reine angew. Math. 136 (1909), 67–178. Google Scholar

About the article

Received: 2014-10-29

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, DOI: https://doi.org/10.1515/crelle-2015-0067.

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