The \(a\)-number stratification on the moduli space of supersingular abelian varieties (Q1877761)

Denote by \(S_g\) the moduli space of principally polarized supersingular Abelian varieties of dimension \(g\) over a field of characteristic \(p> 0\). Recently, the structure of the space \(S_g\) has been intensively studied by several authors, above all by \textit{K.-Z. Li} and \textit{F. Oort} [``Moduli of supersingular abelian varieties'', Lect. Notes Math. 1680 (1998; Zbl 0920.14021)], who introduced various stratifications of \(S_g\) in order to describe the geometry of this classifying object. One particular stratification (due to F. Oort) of \(S_g\) is provided by the so-called ``\(a\)-number'' of an Abelian variety \(X\) over a perfect field. This invariant is defined by \(a(X):= \dim_K\Hom(\alpha_p,X)\), where \(\alpha_p\) is the kernel of the Frobenius map \(F: \mathbb{G}_a\to \mathbb{G}_a\), and the corresponding \(a\)-number stratum \(S_g(a)\) is then defined as a locally closed subscheme of \(S_g\) whose \(K\)-closed points, for any perfect field \(K\) of characteristic \(p> 0\), are characterized by the equation \(a(X) = a\). The main task of the paper under review is to analyze those \(a\)-number strata \(S_g(a)\) more profoundly, especially their irreducible components, and thereby to enhance the previous results of \textit{K.-Z. Li} and \textit{F. Oort} [loc. cit.] in a generalizing manner. The author's principal results presented here establish (1) the connectedness of the Zariski closure of \(S_g(a)\) unless \(a= g\); (2) the pure-dimensionality of \(S_g(a)\) and the dimension formula \[ \dim(S_g(a))= \Biggl[{g^2- a^2+ 1\over 4}\Biggr]; \qquad\text{and} \] (3) formulae for the number of irreducible components of \(S_g(a)\) according to the respective parities of \(g\) and \(a\). Based upon the approach developed by Li and Oort, which is partially reviewed for the convenience of the reader, and using the theory of Dieudonné modules of supersingular Abelian varieties in an essential way, the author adds some crucial new ingredients that lead him to interpret (and to tackle) the strate \(S_g(a)\) as certain period domains. The fine, very creative analysis carried out in this paper is presented in a highly lucid, comprehensive and detailed manner, and the above-mentioned main results must be seen as a major contribution to the moduli theory of supersingular Abelian varieties.