  We study the possible singularities of an $m$-subharmonic function $\varphi$
along a complex submanifold $V$ of a compact K\"ahler manifold, finding a
maximal rate of growth for $\varphi$ which depends only on $m$ and $k$, the
codimension of $V$. When $k < m$, we show that $\varphi$ has at worst log poles
along $V$, and that the strength of these poles is moveover constant along $V$.
This can be thought of as an analogue of Siu's theorem.
