Smooth Approximation of Plurisubharmonic Functions on Almost Complex Manifolds

TL;DR

This paper proves that on almost complex manifolds, J-plurisubharmonic functions can be smoothly approximated from above, extending previous results in complex dimension 2 to higher dimensions by solving an obstacle problem.

Contribution

It establishes the first general method for smooth approximation of J-plurisubharmonic functions on almost complex manifolds in all dimensions.

Findings

Smooth approximation from above for J-plurisubharmonic functions on almost complex manifolds.

Local smooth approximation is possible on J-pseudoconvex domains.

The obstacle problem solution is key to extending the approximation to higher dimensions.

Abstract

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} of smooth, strictly J-plurisubharmonic functions point-wise decreasing down to u. On 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.