Plurisubharmonic functions and positive currents of type (1,1) over almost complex manifolds

If $(X,J)$ is an almost complex manifold, then a function $u$ is said to be plurisubharmonic on $X$ if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is conjectured that plurisubharmonicity is equivalent to the fact that the $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$ is positive, (the $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$ need not be closed here). The conjecture is trivial if $u$ is of class ${\cal C}^2$. The result is elementary in the complex integrable case because the operator $i\partial_{_J}\bar{\partial}_{_J}$ can be written as an operator with constant coefficients in complex coordinates. Hence the positivity of the current is preserved by regularising with usual convolution kernels. This is not possible in the almost complex non integrable case and the proof of the result requires a much more intrinsic study. In this chapter we prove the necessity of the positivity of the $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$. We prove also the sufficiency of the positivity in the particular case of an upper semi-continuous function $f$ which is continuous in the complement of the singular locus $f^{-1}(-\infty)$. For the proof of the sufficiency of the positivity in the general case of a real distribution $u$, we suggest a method depending on a rather delicate regularisation argument introduced by Demailly. This method consists of regularing the function $u$ by means of the flow induced by a Chern connection on the tangent bundle of the almost complex manifold.