Moduli of Abelian Varieties

Volume 195 of the series Progress in Mathematics pp 345-416

A Stratification of a Moduli Space of Abelian Varieties

  • Frans Oort

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In this paper we study the moduli space A0 Fp of polarized abelian varieties of dimensiongin positive characteristic. We construct a stratification of this space. The strata are indexed by isomorphism classes of group schemes killed byp;a polarized abelian variety (X, A) has its moduli point in a certain stratum if X [p] belongs to the isomorphism class given by a certain discrete invariant. We define these invariants by a numerical property of a filtration ofN= X [p].Passing from one stratum to a stratum in its boundary feels like “degenerating the p-structure”. The fact that these strata are all quasi-affine allows us to keep going in this process until we arrive at the unique zero-dimensional stratum, the superspecial locus. One can formulate this idea by saying that the ordinary locus has several “boundaries”, one where the abelian variety degenerates, one where the p-structure “becomes more special” (and an analogous idea for all non-zero-dimensional strata). This phenomenon, non-present in this form in characteristic zero, but available and powerful in positive characteristic, is ex-pected to have many applications.