I feel confused about a point in this very short paper. On the top of page 3, it is claimed that:

*If $S$ is a totally real submanifold in a compact almost complex manifold $(X,J)$, then any function $\rho\ge 0$ near $S$, vanishing on $S$ (and non-degenerate transversally to $S$), must be strictly $J$-plurisubharmonic in a neighborhood of $S$.*

We say a function $\rho$ is *strictly* $J$-plurisubharmonic if $dd^J \rho >0$, where $d^J \rho = -d\rho \circ J$. For a two-form $\theta$ we write $\theta>0$ if for any $v\neq 0$ we have $\theta(v,Jv)>0$.

The author's explanation is very sketchy and I get confused. It seems that it is sort of standard result but I fail to find any reference. I think experts in MO should know this very well.

Thank you!