Cycles on the moduli space of abelian varieties. (English) Zbl 0974.14031

Faber, Carel (ed.) et al., Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Braunschweig: Vieweg. Aspects Math. E33, 65-89 (1999).
From the paper: The author presents a number of results on cycles on the moduli space \({\mathcal A}_g\) of principally polarized abelian varieties of dimension \(g\). Our results include:
a description of the tautological subring of the Chow ring of \({\mathcal A}_g\), i.e. of the subring generated by the Chern classes \(\lambda_i\) of the Hodge bundle \(\mathbb E\);
a formula for the top Chern class \(\lambda_g\) of the Hodge bundle and a bound for the order of the torsion of this class;
a description of the Ekedahl-Oort stratification [cf. F. Oort, same volume, Aspects Math. E33, 47–64 (1999; Zbl 0974.14029)] of \({\mathcal A}_g\otimes \mathbb F_p\) in terms of degeneracy loci of a map between flag bundles;
the description of the Chow classes of the strata of this stratification. This includes as special cases formulas for the classes of loci like \(p\)-rank \(\leq f\) locus or \(a\)-number \(\geq a\) locus. Such formulas generalize the classical formula of Deuring for the number of supersingular elliptic curves;
the irreducibility of the locus \(T_a\) of abelian varieties of \(a\)-number \(\geq a\) for \(a<g\);
a computation of this stratification for the Jacobian of hyperelliptic curves of 2-rank 0 in characteristic 2;
a formula for the class of the supersingular locus for low genera.
The results on the tautological ring are the author’s own work, the results on the torsion of \(\lambda_g\) and on the cycle classes of the Ekedahl-Oort stratification are joint work with Torsten Ekedahl and some of the results on curves are joint work with Carel Faber.
For the entire collection see [Zbl 0933.00030].


14K10 Algebraic moduli of abelian varieties, classification
14C25 Algebraic cycles
14C05 Parametrization (Chow and Hilbert schemes)


Zbl 0974.14029
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