2 00 8 Plurisubharmonicity in a General Geometric Context

Recently the authors have explored new concepts of plurisubharmonic-ity and pseudoconvexity, with much of the attendant analysis, in the context of calibrated manifolds. Here a much broader extension is made. This development covers a wide variety of geometric situations , including, for example, Lagrangian plurisubhamonicity and con-vexity. It also applies in a number of non-geometric situations. Results include: fundamental properties of P +-plurisubharmonic functions , plurisubharmonic distributions and regularity, P +-convex domains and P +-convex boundaries, topological restrictions on and construction of such domains, continuity of upper envelopes, and solutions of the Dirichlet problem for related Monge-Ampère-type equations.