For families of continuous plurisubharmonic functions we show that, in a local sense, separately bounded above implies bounded above.

We continue the study in [2] in the setting of weighted pluripotential theory arising from polynomials associated to a convex body P in (R+)d. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of P-pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge–Ampère equation in an appropriate finite energy class. This is achieved using a variational approach.

The concept of a selfdual connection on a fourdimensional Riemannian manifold is generalized to the 4ndimensional case of any quaternionic Khler manifold. The generalized selfdual connections are minima of a modified YangMills functional. It is shown that our definitions give a correct framework for a mapping theory of quaternionic Khler manifolds. The mapping theory is closely related to the construction of YangMills fields on such manifolds. Some monopolelike equations are discussed.

It is proved that dicritical singularities of real analytic Levi-flat sets coincide with the set of Segre degenerate points.

A generalization of a result of Wermer concerning the existence of polynomial hulls without analytic discs is presented. As a consequence it is shown that there exists a Cantor set X in C3 whose polynomial hull is strictly larger than X but contains no analytic discs.