Smooth Approximation of Plurisubharmonic Functions on Almost Complex Manifolds

This note establishes smooth approximation from above for Jplurisubharmonic functions on an almost complex manifold (X, J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic function on X. Then there exists a sequence uj ∈ C∞(X) of smooth strictly Jplurisubharmonic functions point-wise decreasing down to u. In any almost complex manifold (X, J) each point has a fundamental neighborhood system of J-pseudoconvex domains, and so the theorem above establishes local smooth approximation on X. This result was proved in complex dimension 2 by the third author, who also showed that the result would hold in general dimensions if a parallel result for continuous approximation were known. This paper establishes the required step by solving the obstacle problem.