Simon Cooper: Intersection Theory on Zip Period Maps

Abstract.

This thesis consists of four papers, all motivated by questions about intersection theory on Shimura varieties in positive characteristic. The connection with intersection theory of flag varieties, made using the stack of \(G\)-Zips of type \(\mu\), is explored throughout. More generally, we work in the setting of intersection theory on spaces \(X\) admitting morphisms to the stack of \(G\)-Zips of type \(\mu\). These morphisms are termed 'zip period maps' in Paper III. The fundamental example of such an \(X\) is the special fibre of an integral canonical model of a Shimura variety of Hodge-type. Moreover, there is a notion of 'tautological ring' for any (smooth) zip period map which gives the usual tautological ring in the case of Shimura varieties.

In Paper I the tautological ring of a Hilbert modular variety at an unramified prime is computed. The method generalises van der Geer's approach from the Siegel case and makes use of the properness of the non-maximal Ekedahl-Oort strata closures in this setting.

The pushforward map in the Chow ring between Siegel flag varieties is computed in Paper II. Siegel flag varieties are projective varieties which are quotients of the symplectic group. They appear as the compact dual of the Siegel upper half plane. A conjecture exploring the connection between classes in Chow rings of flag varieties and classes in tautological rings of Shimura varieties is presented. The computation contained in this paper can be viewed as very basic evidence for this conjecture.

In Paper III we develop various conjectures related to positivity in the tautological ring of a zip period map. The notion of strata-effective classes is introduced. Several conjectures are presented regarding classes which we expect to be strata-effective. These are proved in many cases, including for Hilbert modular varieties, which are more accessible for various group-theoretic reasons. A connection between strata-effectivity and the Cone Conjecture of Goldring-Koskivirta is developed and provides examples of tautological and effective classes which nevertheless fail to be strata-effective.

In Paper IV we compute the Grothendieck group of the stack of \(G\)-Zips of type \(\mu\) (as a ring) in the case where the derived group of \(G\) is simply connected.