Minimum co-degree condition for perfect matchings in k-partite k-graphs

Let $H$ be a $k$-partite $k$-graph with $n$ vertices in each partition class, and let ${\delta }_{k-1}(H)$ denote the minimum co-degree of $H$. We characterize those $H$ with ${\delta }_{k-1}(H)\ge n\mathrm{/}2$ and with no perfect matching. As a consequence we give an affirmative answer to the following question of R"odl and Ruci'nski: If $k$ is even or $n\equiv ̸2(\mathrm{m}\mathrm{o}\mathrm{d}4)$, does ${\delta }_{k-1}(H)\ge n\mathrm{/}2$ imply that $H$ has a perfect matching? We also give an example indicating that it is not sufficient to impose this degree bound on only two types of $(k-1)$-sets.