Some intersection numbers of divisors on toroidal compactifications of $\mathcal {A}_g$
Authors:
C. Erdenberger, S. Grushevsky and K. Hulek
Journal:
J. Algebraic Geom. 19 (2010), 99-132
DOI:
https://doi.org/10.1090/S1056-3911-09-00512-8
Published electronically:
April 21, 2009
MathSciNet review:
2551758
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Abstract |
References |
Additional Information
Abstract: We study the top intersection numbers of the boundary and Hodge class divisors on toroidal compactifications of the moduli space $\mathcal {A}_g$ of principally polarized abelian varieties and compute those numbers that live away from the stratum which lies over the closure of $\mathcal {A}_{g-3}$ in the Satake compactification.
References
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References
- V. Alexeev, Complete moduli in the presence of semiabelian group action. Ann. of Math. (2) 155 (2002) 3, 611–708. MR 1923963 (2003g:14059)
- Ch. Birkenhake, H. Lange, Complex abelian varieties: second, augmented edition. Grundlehren der mathematischen Wissenschaften 302, Springer-Verlag, Berlin 2004. MR 2062673 (2005c:14001)
- C. Erdenberger, S. Grushevsky, K. Hulek, Intersection theory of toroidal compactifications of $\mathcal {A}_4$, Bull. London Math. Soc. 38 (2006), 396–400. MR 2239033 (2007b:14098)
- H. Esnault, E. Viehweg, Chern classes of Gauss-Manin bundles of weight 1 vanish, $K$-Theory 26 (2002), 287–305. MR 1936356 (2003i:14059)
- G. Faltings, C.-L. Chai, Degeneration of abelian varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete 22. Springer-Verlag, Berlin, 1990. MR 1083353 (92d:14036)
- W. Fulton, H. Gillet Riemann-Roch for general algebraic varieties, Bulletin de la SMF 11 (1983), 287-300. MR 735307 (85h:14010)
- W. Fulton, Intersection theory. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
- G. van der Geer, The Chow ring of the moduli space of abelian threefolds. J. Algebraic Geom. 7 (1998), 753–770. MR 1642753 (2000b:14009)
- G. van der Geer, Cycles on the moduli space of abelian varieties. Moduli of curves and abelian varieties, 65–89, Aspects Math., E33, Vieweg, Braunschweig, 1999. MR 1722539 (2001b:14009)
- R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA 1989. MR 1397498 (97d:68003)
- S. Grushevsky, D. Lehavi, Some intersections in the Poincaré bundle and the universal theta divisor on $\overline {\mathcal {A}_g}$, arXiv:math.AG/0512530v2.
- K. Hulek, Nef divisors on moduli spaces of abelian varieties, Complex analysis and algebraic geometry, 255–274, de Gruyter, Berlin, 2000. MR 1760880 (2001d:14046)
- K. Hulek, G. K. Sankaran, The nef cone of toroidal compactifications of $\mathcal {A}_4$. Proc. London Math. Soc. 88 (2004), 659–704. MR 2044053 (2005a:14061)
- Hulek, K., Weintraub, S.: The principal degenerations of abelian surfaces and their polarisations, Math. Ann. 286 (1990), 281–307. MR 1032935 (91e:14042)
- Maple$^{\mathrm {TM}}$ 10 software, copyright by Maplesoft
- D. Mumford, On the Kodaira dimension of the Siegel modular variety. In: Algebraic geometry–open problems (Ravello, 1982), 348–375, Lecture Notes in Math. 997, Springer, Berlin, 1983. MR 714757 (85d:14061)
- D. Mumford, Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and Geometry II, Progress in Mathematics 36 (1983), 271–328. MR 717614 (85j:14046)
- M. Olsson, Compactifying moduli spaces of abelian varieties. Lecture Notes in Mathematics, 1958. Springer-Verlag, Berlin, 2008. MR 2446415
- N. Shepherd-Barron, Perfect forms and the moduli space of abelian varieties, Invent. Math. 163 (2006), 25–45. MR 2208417 (2007e:14070)
- R. Tsushima, A formula for the dimension of spaces of Siegel modular cusp forms. Am. J. Math. 102 (1980), 937–977. MR 590640 (81m:10050)
- G.F. 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. 133 (1908), 79–178.
- G.F. 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.
- G.F. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les paralléloèdres primitifs. Seconde partie. Domaines de formes quadratiques correspondant aux différents types de paralléloèdres primitifs, J. Reine Angew. Math. 136 (1909), 67–178.
Additional Information
C. Erdenberger
Affiliation:
Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
Email:
erdenber@math.uni-hannover.de
S. Grushevsky
Affiliation:
Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544
MR Author ID:
192264
Email:
sam@math.princeton.edu
K. Hulek
Affiliation:
Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
MR Author ID:
89705
Email:
hulek@math.uni-hannover.de
Received by editor(s):
July 17, 2007
Received by editor(s) in revised form:
January 24, 2008
Published electronically:
April 21, 2009
Additional Notes:
The second author’s research is supported in part by the National Science Foundation under the grant DMS-05-55867. The third author’s research is supported in part by DFG grant Hu 337/5-3.
