Smooth approximation of plurisubharmonic functions on almost complex manifolds (Q343153)

Let \((X, J)\) be a smooth almost complex manifold. An upper semicontinuous function on \(X\) is called \(J\)-plurisubharmonic if its restriction to any \(J\)-holomorphic curve in \(X\) is subharmonic. If this function is of class \(C^2\) and \(i\partial\bar{\partial } >0\) then it is strictly \(J\)-plurisubharmonic. The manifold \(X\) is \(J\)-pseudoconvex if it admits a strictly \(J\)-plurisubharmonic exhaustion function. For such a manifold, consider a \(J\)-plurisubharmonic function \(u\) on \(X\). The authors show that one can find a decreasing sequence of \(C^{\infty }\) strictly \(J\)-plurisubharmonic functions converging to \(u\) on \(X\). Previously, the result was only known in dimension \(2\) [\textit{S. Pliś}, C. R., Math., Acad. Sci. Paris 353, No. 1, 17--19 (2015; Zbl 1310.32036)].