Oort’s conjecture for $A_{g} \otimes {\mathbb {C}}$

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

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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 $g$ has codimension strictly greater than $g$, in characteristic zero, for $g \geq 3$.

- Theodor Bröcker and Tammo tom Dieck,
*Representations of compact Lie groups*, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR**781344** - 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** - E. Colombo and G. P. Pirola,
*Some density results for curves with nonsimple Jacobians*, Math. Ann.**288**(1990), no. 1, 161–178. MR**1070930**, DOI https://doi.org/10.1007/BF01444527 - 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]EV02 H. Esnault and E. Viehweg, - 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** - 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** - 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**, DOI https://doi.org/10.1307/mmj/1030132716 - 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** - Phillip Griffiths (ed.),
*Topics in transcendental algebraic geometry*, Annals of Mathematics Studies, vol. 106, Princeton University Press, Princeton, NJ, 1984. MR**756842** - Phillip Griffiths and Joseph Harris,
*Principles of algebraic geometry*, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR**507725** - 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**, DOI https://doi.org/10.1007/978-3-322-90172-9_4 - 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**, DOI https://doi.org/10.1007/978-3-322-90172-9_1 - 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**, DOI https://doi.org/10.1007/s002080050146 - 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** - George R. Kempf,
*Complex abelian varieties and theta functions*, Universitext, Springer-Verlag, Berlin, 1991. MR**1109495**
[Koblitz75]Koblitz75 N. Koblitz, - Jean-Pierre Serre,
*Lie algebras and Lie groups*, W. A. Benjamin, Inc., New York-Amsterdam, 1965. Lectures given at Harvard University, 1964. MR**0218496** - Robert J. Zimmer,
*Ergodic theory and semisimple groups*, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR**776417**

*Chern classes of Gauss-Manin bundles of weight $1$ vanish*, preprint math.AG/020103, 2002.

*$p$-adic variation of the zeta-function over families of varieties defined over finite fields*, Compos. Math.

**31**(1975), 119–218.

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Additional Information

**Sean Keel**

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

MR Author ID:
289025

Email:
keel@math.utexas.edu

**Lorenzo Sadun**

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

Email:
sadun@math.utexas.edu

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