Automorphic vector bundles with global sections on G-ZipZ-schemes.

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of G-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type A1n, C2, and Fp-split groups of type A2 (this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular 3-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type. [ABSTRACT FROM AUTHOR]