Positivity of automorphic vector bundles on unitary Shimura varieties

Let $X$ be the special fiber of a unitary Shimura variety of hyperspecial level at a prime $p$ inert in the totally real field $F$. Let $Y\to X$ be the associated flag space. For every $L$-dominant weight $\lambda$, let ${\mathcal{L}}_Y(\lambda )$ denote the corresponding automorphic line bundle. We give an explicit necessary and sufficient criterion, in terms of the signature data and the coordinates of $\lambda$, for the ampleness of ${\mathcal{L}}_Y(\lambda )$. %, which effectively detects the coherent cohomology of automorphic vector bundles on $X$. The criterion generalizes the known ample cone for Hilbert modular and $U(2)$-Shimura varieties. The proof develops the machinery of the description of certain Ekedahl–Oort strata, a geometric Jacquet–Langlands correspondence between strata of unitary Shimura varieties with different signatures, and the construction of stratum Hasse invariants, and introduced a way to systematically deal with combinatorical data in the higher rank case.