Oort's conjecture for

Authors:
Sean Keel and Lorenzo Sadun

Journal:
J. Amer. Math. Soc. **16** (2003), 887-900

MSC (2000):
Primary 14K10

DOI:
https://doi.org/10.1090/S0894-0347-03-00431-4

Published electronically:
May 30, 2003

MathSciNet review:
1992828

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the conjecture of Oort that a compact subvariety of the moduli space of principally polarized Abelian varieties of genus has codimension strictly greater than , in characteristic zero, for .

**[BD85]**Theodor Bröcker and Tammo tom Dieck,*Representations of compact Lie groups*, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR**781344****[BT82]**Raoul Bott and Loring W. Tu,*Differential forms in algebraic topology*, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR**658304****[CP90]**E. Colombo and G. P. Pirola,*Some density results for curves with nonsimple Jacobians*, Math. Ann.**288**(1990), no. 1, 161–178. MR**1070930**, https://doi.org/10.1007/BF01444527**[Diaz87]**Steven Diaz,*Complete subvarieties of the moduli space of smooth curves*, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 77–81. MR**927950****[EV02]**H. Esnault and E. Viehweg,*Chern classes of Gauss-Manin bundles of weight**vanish*, preprint math.AG/020103, 2002.**[Faber99]**Carel Faber,*A conjectural description of the tautological ring of the moduli space of curves*, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 109–129. MR**1722541****[FL99]**Carel Faber and Eduard Looijenga,*Remarks on moduli of curves*, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 23–45. MR**1722537****[FP00]**C. Faber and R. Pandharipande,*Logarithmic series and Hodge integrals in the tautological ring*, Michigan Math. J.**48**(2000), 215–252. With an appendix by Don Zagier; Dedicated to William Fulton on the occasion of his 60th birthday. MR**1786488**, https://doi.org/10.1307/mmj/1030132716**[Fulton84]**William Fulton,*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR**732620****[Griffiths84]**Phillip Griffiths (ed.),*Topics in transcendental algebraic geometry*, Annals of Mathematics Studies, vol. 106, Princeton University Press, Princeton, NJ, 1984. MR**756842****[GH78]**Phillip Griffiths and Joseph Harris,*Principles of algebraic geometry*, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR**507725****[G99]**Gerard van der Geer,*Cycles on the moduli space of abelian varieties*, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 65–89. MR**1722539**, https://doi.org/10.1007/978-3-322-90172-9_4**[GO99]**Gerard van der Geer and Frans Oort,*Moduli of abelian varieties: a short introduction and survey*, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 1–21. MR**1722536**, https://doi.org/10.1007/978-3-322-90172-9_1**[Izadi98]**E. Izadi,*Density and completeness of subvarieties of moduli spaces of curves or abelian varieties*, Math. Ann.**310**(1998), no. 2, 221–233. MR**1602079**, https://doi.org/10.1007/s002080050146**[Mumford83]**David Mumford,*Towards an enumerative geometry of the moduli space of curves*, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328. MR**717614****[Kempf91]**George R. Kempf,*Complex abelian varieties and theta functions*, Universitext, Springer-Verlag, Berlin, 1991. MR**1109495****[Koblitz75]**Neal Koblitz,*𝑝-adic variation of the zeta-function over families of varieties defined over finite fields*, Compositio Math.**31**(1975), no. 2, 119–218. MR**414557****[Serre65]**Jean-Pierre Serre,*Lie algebras and Lie groups*, Lectures given at Harvard University, vol. 1964, W. A. Benjamin, Inc., New York-Amsterdam, 1965. MR**0218496****[Zimmer84]**Robert J. Zimmer,*Ergodic theory and semisimple groups*, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR**776417**

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2000):
14K10

Retrieve articles in all journals with MSC (2000): 14K10

Additional Information

**Sean Keel**

Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78712

Email:
keel@math.utexas.edu

**Lorenzo Sadun**

Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas, 78712

Email:
sadun@math.utexas.edu

DOI:
https://doi.org/10.1090/S0894-0347-03-00431-4

Received by editor(s):
May 1, 2002

Published electronically:
May 30, 2003

Additional Notes:
The first author was partially supported by NSF grant DMS-9988874

The second author was partially supported by Texas ARP grant 003658-152

Article copyright:
© Copyright 2003
American Mathematical Society