Connected components of special cycles on Shimura varieties

TL;DR

This paper investigates the structure of special cycles on GSpin Shimura varieties, establishing irreducibility of certain fibers and applying these results to moduli of K3 surfaces and modularity of cycle generating series.

Contribution

It introduces new methods for analyzing irreducible components of special cycles and applies them to K3 surface moduli and modularity conjectures.

Findings

Irreducibility of special fibers in certain cases

Application to moduli of polarized K3 surfaces

Results on modularity of higher codimension cycles

Abstract

I use methods of Chai-Hida and ordinary $p$-Hecke correspondences to study the set of irreducible components of special fibers of special cycles of sufficiently low codimension in integral models of GSpin Shimura varieties, and apply this to prove irreducibility results for the special fibers of the moduli of polarized K3 surfaces. These results are also applied in joint work with Howard on the modularity of generating series of higher codimension cycles on GSpin Shimura varieties.