On the singularities and the Kodaira dimension of unitary Shimura varieties

Abstract

We study the geometry of Shimura varieties associated with a Hermitian form of signature $(p,q)$ over an imaginary quadratic field $E$, where $2\leq p\leq q$. We prove that when $p\geq w_E$ and $(p,q)\neq(2,2),(2,3)$, where $w_E:=\#\mathscr{O}_E^\times$, there exists a toroidal compactification with at worst canonical singularities. As an application, combining this singularity analysis with Arthur's multiplicity formula, we prove that only finitely many pairs $(p,q)$ satisfying the above conditions and $p+q\equiv1\pmod{w_E}$ give rise to unitary Shimura varieties that are not of general type. Our method also improves the singularity bound of Gritsenko--Hulek--Sankaran (Invent.\ Math., 2007) for $\mathrm{O}^+(2,n)$ to the range $n\geq6$ and shows that this bound is sharp.