On the singularities and the Kodaira dimension of unitary Shimura varieties

We study the geometry of Shimura varieties associated with a Hermitian form of signature $(p,q)$ over an imaginary quadratic field $E$, where $2\le p\le q$. We prove that when $p\ge w_E$ and $(p,q)\mathrm{\ne }(2,2),(2,3)$, where $w_E:=\mathrm{\#}{\mathcal{O}}_E^{\times }$, 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 $p+q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}{w}_E)$ 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 $n\ge 6$ and shows that this bound is sharp.