  We study the decomposition matrices for the unipotent $\ell$-blocks of finite
special unitary groups SU$_n(q)$ for unitary primes $\ell$ larger than $n$. Up
to very few unknown entries, we give a complete solution for $n=2,\ldots,10$.
We also prove a general result for two-column partitions when $\ell$ divides
$q+1$. This is achieved using projective modules coming from the $\ell$-adic
cohomology of Deligne--Lusztig varieties.
