Lelong Numbers of $m$-Subharmonic Functions Along Submanifolds

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.