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

van der Geer, Gerard

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].

MSC:

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

Keywords:

stratification cycles; Chow ring; moduli spaces of principally polarized varieties

Citations:

Zbl 0974.14029