Positivity of automorphic vector bundles on unitary Shimura varieties

Let X be the special fiber of a unitary Shimura variety of hyperspecial level. We establish a sharp criterion for the ampleness of automorphic line bundles LY(λ) over the flag space Y, which effectively detects the coherent cohomology of automorphic vector bundles on X. Our proof is based on a careful analyse of the stratification of X and Y, along with the morphisms between these strata. Additionally, we construct certain strata Hasse invariants, and establish correspondences between specific strata of different unitary Shimura varieties and their respective flag spaces.