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

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