Supersingular loci of some Shimura varieties for CM unitary groups

In this note, we study Shimura varieties for the groups $\mathrm{G}\mathrm{U}(V)$, where $V$ is a Hermitian space relative to a CM extension $E\mathrm{/}{E}^{+}$. We give a description of the supersingular locus of the fiber at a prime $\nu$ over $p$ of such a Shimura variety, under the assumptions that $$\mathrm{d}\mathrm{i}\mathrm{m}$_E(V) \leq 4$andthattheprime$p$splitscompletelyin$E^+$andisunramifiedin$E$.