# Automatic plurisubharmonicity for a non-negative function

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!

• Can you give the precise definition of non-degenerate and strictly plurisubharmonic? You are talking about function $\rho$, while give the definition for a form. And at which points should the inequality $\theta(v,Jv)>0$ hold? On $S$ or in a neighborhood of $S$? – Alexandre Eremenko Dec 16 '18 at 6:36
• @AlexandreEremenko Very sorry. I post the question in a hurry and didn't notice that my question is very unclear. Now I have edited the question. However I still do not know the precise meaning of 'non-degenerate' as the author didn't write an explicit definition either. But according to the context, the non-degeneracy should mean (I am not completely sure) $\rho=O(|y|^2)$ where $|y|$ is roughly the distance to $S$. – Hang Dec 16 '18 at 15:39
• you still did not state WHERE $dd^J\rho(x)>0$ should hold. On $S$? – Alexandre Eremenko Dec 16 '18 at 15:46
• $\rho$ is defined near $S$ and so $dd^J\rho>0$ in a neighborhood of $S$ not just on $S$. – Hang Dec 16 '18 at 15:46
• Then this is not true. In dimension one, take $S=R$, $\rho(z)=|\Im z|$. – Alexandre Eremenko Dec 16 '18 at 15:48