Positivity of automorphic vector bundles on unitary Shimura varieties

TL;DR

The paper establishes a sharp criterion for the ampleness of automorphic line bundles on the flag space over the special fiber of a unitary Shimura variety of hyperspecial level, linking this positivity to the coherent cohomology of automorphic vector bundles on the original variety. It develops a rich toolkit based on the stack of G-zips, Ekedahl–Oort stratification, and stratified flag spaces, and introduces strata Hasse invariants to control intersection numbers. A key innovation is a geometric correspondence between strata across different unitary Shimura varieties, enabling transfer of positivity data, and a detailed analysis via Dieudonné theory and $F,V$-chains to relate local and global line bundles. The authors prove a precise Ampleness Criterion that depends on the essential degree and the weight components, and they provide a nefness-inequality framework that yields vanishing results for coherent cohomology and related automorphic applications. The work advances the understanding of mod $p$ geometry of unitary Shimura varieties and offers tools potentially extensible to broader Hodge-type cases.

Abstract

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 $\mathcal{L}_Y(λ)$ 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.

Theorems & Definitions (88)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Proposition 1.9: Theorem \ref{['Description of strata']}
• Remark 2.1