Smooth approximation of plurisubharmonic functions on almost complex manifolds

This note establishes smooth approximation from above for J-plurisubharmonic 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 u_j∈ C^∞(X) of smooth strictly J-plurisubharmonic 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.