Ampleness of Automorphic Line Bundles on $U(2)$ Shimura Varieties

TL;DR

This paper proves a conjecture about the ampleness of automorphic line bundles on $U(2)$ Shimura varieties, establishing precise conditions on coefficients for ampleness on the special fiber, extending to Hilbert modular varieties.

Contribution

It proves the conjecture for $U(2)$ Shimura varieties regarding ampleness conditions of automorphic line bundles on the special fiber.

Findings

Automorphic line bundle $ ext{L}$ is ample on the generic fiber if coefficients are positive.

The paper confirms the conjecture relating coefficients to ampleness on the special fiber for $U(2)$ Shimura varieties.

Results extend to Hilbert modular varieties, providing new insights into their geometric properties.

Abstract

Let $F$ be a totally real field in which $p$ is unramfied and let $S$ denote the integral model of the Hilbert modular variety with good reduction at $p$. Consider the usual automorphic line bundle $\mathcal{L}$ over $S$. On the generic fiber, it is well known that $\mathcal{L}$ is ample if and only if all the coefficients are positive. On the special fiber, it is conjectured in \citep{Tian-Xiao} that $\mathcal{L}$ is ample if and only if the coefficients satisfy certain inequalities. We prove this conjecture for $U(2)$ Shimura varieties in this paper and deduce a similar statement for Hilbert modular varieties from this.