Continuous approximation of quasiplurisubharmonic functions (Q2786816)

The paper under review deals with the regularization problem of quasiplurisubharmonic functions. The main result of the paper is the following theorem. Suppose that \(X\) is a compact Kähler manifold and \(\alpha\in H^{1,1}(X,{\mathbb R})\) is a big cohomology class for which the polar locus coincides with the unbounded locus. If \(\theta\in\alpha\) is a smooth form then any \(\theta\)-plurisubharmonic function on \(X\) is the limit of a decreasing sequence of exponentially continuous \(\theta\)-plurisubharmonic functions on \(X\) with minimal singularities.NEWLINENEWLINENote that the existence of an exponentially continuous \(\theta\)-plurisubharmonic function on \(X\) with minimal singularities implies the equality of the polar and the unbounded loci of \(\alpha\), hence this hypothesis is necessary in the above theorem. As a corollary to the theorem the authors show that on a compact normal Kähler space \((V,\omega_V)\) any \(\omega_V\)-plurisubharmonic function is the limit of a decreasing sequence of smooth \(\omega_V\)-plurisubharmonic functions.NEWLINENEWLINEThe proof of the main result relies on regularization techniques of Demailly, as well as on earlier work of the authors on solving the complex Monge-Ampère equation in big cohomology classes and on viscosity solutions for the complex Monge-Ampère equation. In particular it is shown here that the viscosity comparison principle holds for big cohomology classes whose polar and unbounded loci are equal.NEWLINENEWLINEFor the entire collection see [Zbl 1322.53004].