The supersingular locus in Siegel modular varieties with Iwahori level structure (Q425150)

Let \(g \geq 1\) be an integer, \(p\) be a prime number, and \(k\) be an algebraic closure of \(\mathbb{F}_p\). Denote by \(\mathcal{A}_g\) the moduli space of principally polarized abelian varieties of dimension \(g\) over \(k\), and \(\mathcal{A}_I\) the moduli space of abelian varieties of dimension g with Iwahori level structure at \(p\), that is the space of chains \(A_0 \to A_1 \to \cdots \to A_g\) of isogenies of order \(p\) satisfying a pull-back condition that concerns principal polarizations of \(A_0\) and \(A_g\). Let \(\mathcal{S}_I\) be supersingular locus inside \(\mathcal{A}_I\). This paper proves the following results: Theorem: 1) If \(g\) is even, then \(\mathrm{dim}\;\mathcal{S}_I = g^2/2\); if \(g\) is odd, then \[ \frac{g(g-1)}{2} \leq \mathrm{dim}\; \mathcal{S}_I \leq \frac{(g+1)(g-1)}{2}. \] 2) Suppose \(g\) is even. Then every top-dimensional irreducible component of \(\mathcal{S}_I\) is isomorphic to the full flag variety of the group \(\mathrm{Sp}_{2\frac{g}{2}} \times \mathrm{Sp}_{2\frac{g}{2}}\) (over \(k\)). 3) Suppose \(g\) is odd. Then \(\mathcal{S}_I\) has irreducible components which are isomorphic to the flag variety of \(\mathrm{SL}_g\) (over \(k\)). (So these components are of dimension \(g(g-1)/2\).) The main theorem is obtained by proving the following important facts. Let \(\pi:\mathcal{A}_I \to \mathcal{A}_g\) be the natural projection. Using relative position of chains, the authors proved that each Kottwitz-Rapoport (KR) stratum is mapped to a union of Ekedahl-Oort (EO) strata. Then they proved that all KR strata in \(\mathcal{A}_I\) are quasi-affine. This is achieved by studying ample line bundles on \(\mathcal{A}_I\). Next they prove that all non-superspecial KR strata are connected, following the strategy of \textit{T. Ekedahl} and \textit{G. van der Geer} in their paper [Progr. Math. 269, 567--636 (2009; Zbl 1200.14089)]. In the end, dimension of the \(p\)-rank \(0\) locus has been calculated with quite sophisticated computations. Combining this with Proposition 4.6 in the authors' previous paper [J. Inst. Math. Jussieu 9, No. 2, 357--390 (2010; Zbl 1191.14028)], the main theorem is proved.