Stefan Reppen: Hasse invariants on Shimura varieties and moduli of G-bundles

Abstract.

This thesis consists of four papers, referred to in the text as Paper I, II, III and IV, respectively. In Paper I and II, we study the classical Hasse invariant on the geometric special fiber of several Hodge and abelian-type Shimura varieties. We explicitly compute the order of vanishing of the Hasse invariant and describe it in terms of the Bruhat and Ekedahl-Oort stratification of the variety. We also compute the conjugate line position (referred to as the \(a\)-number by van der Geer and Katsura, respectively Oort) and thus show that it agrees with the order of the Hasse invariant. This equality was obtained by Ogus for certain families of Calabi-Yau varieties mod \(p\), and we thus dub it as Ogus' principle. We give a group-theoretical generalization of this principle via the theory of \(G\)-zips, and argue that this theory is a suitable framework within which to understand the principle.

Paper III and IV concern (moduli of) \(G\)-bundles over smooth projective curves over an algebraically closed field, for \(G\) a reductive group over the base field. In Paper III we introduce the notion of essentially finite (EF) \(G\)-bundles, which generalizes the notion of EF vector bundles, studied initially by Weil and more generally by Nori. We state some elementary characterizations of such bundles, and we prove that they are semistable of torsion degree. Let \(M_G^{\text{ef}}\) denote the subset of EF bundles in the moduli space of degree \(0\) semistable \(G\)-bundles. We show that in characteristic \(0\), \(M_G^{\text{ef}}\) is not dense if \(G\) of semisimple rank 1 and the curve has genus \(g\geq 2\). This is in contrast to the case of line bundles in arbitrary characteristic, and vector bundles of arbitrary rank in positive characteristic.

In Paper IV we first study the moduli stack \(\operatorname{Bun}_{\mathscr{G}}\) of \(\mathscr{G}\)-bundles for nonconnected reductive group schemes \(\mathscr{G}\) over the curves. We prove that the semistable locus of \(\operatorname{Bun}_{\mathscr{G}}\) admits a projective good moduli space. To this end we prove a "decomposition theorem" stating that \(\operatorname{Bun}_{\mathscr{G}}\) is a finite disjoint union of substacks \(\mathscr{X}_i\), each of which admits a finite torsor \(\operatorname{Bun}_{\mathscr{G}_i}\to \mathscr{X}_i\) for some connected reductive group schemes \(\mathscr{G}_i\). We expand the nondensity result in Paper III and show that if the base field has characteristic \(0\) and the curve has genus \(g\geq 2\), then \(\dim \overline{M_G^{\text{ef}}}\leq g\operatorname{rk}(G)\). In particular, \(M_G^{\text{ef}}\) is not dense unless \(G\) is a torus. To this end we give a generalization of Jordan's classical result on finite subgroups of \(\mathrm{GL}(n)\).