We prove that a compact subset of full measure on a generic submanifold of an almost complex manifold is not a pluripolar set. Several related results on boundary behavior of plurisubharmonic functions are established. Our approach is based on gluing a family of complex discs to a generic manifold along a boundary arc and could admit further applications.

We use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge–Ampére type equations on compact Kähler manifolds.

This article establishes a sufficient condition for Kobayashi hyperbolicity of unbounded domains in terms of curvature. Specifically, when $\Omega \,\subset \,{{\mathbb{C}}^{n}}$ corresponds to a sub-level set of a smooth, real-valued function Ψ such that the form $\omega \,=\,\mathbf{i}\partial \bar{\partial }\Psi $ is Kähler and has bounded curvature outside a bounded subset, then this domain admits a hermitian metric of strictly negative holomorphic sectional curvature.

We reduce the openness conjecture of Demailly and Kollár on the singularities of plurisubharmonic functions to a purely algebraic statement.

In this note we study the convergence of sequences of Monge–Ampère measures $\{{{(d{{d}^{c}}{{u}_{s}})}^{n}}\}$, where $\{{{u}_{s}}\}$ is a given sequence of plurisubharmonic functions, converging in capacity.

We give a new characterization of Hardy martingale cotype property of complex quasi-Banach space by using the existence of a kind of plurisubharmonic functions. We also characterize the best constants of Hardy martingale inequalities with values in the complex quasi-Banach space.