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!

  • 1
    $\begingroup$ 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 $S$? $\endgroup$ – Alexandre Eremenko Dec 16 '18 at 6:36
  • $\begingroup$ @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 $S$. $\endgroup$ – Hang Dec 16 '18 at 15:39
  • $\begingroup$ you still did not state WHERE $dd^J\rho(x)>0$ should hold. On $S$? $\endgroup$ – Alexandre Eremenko Dec 16 '18 at 15:46
  • $\begingroup$ $\rho$ is defined near $S$ and so $dd^J\rho>0$ in a neighborhood of $S$ not just on $S$. $\endgroup$ – Hang Dec 16 '18 at 15:46
  • $\begingroup$ Then this is not true. In dimension one, take $S=R$, $\rho(z)=|\Im z|$. $\endgroup$ – Alexandre Eremenko Dec 16 '18 at 15:48

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