Continuous approximation of quasiplurisubharmonic functions.

International audienceThe 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 K\"ahler 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.Note 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 K\"ahler space $(V,\omega_V)$ any $\omega_V$-plurisubharmonic function is the limit of a decreasing sequence of smooth $\omega_V$-plurisubharmonic functions.The 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-Amp\`ere equation in big cohomology classes and on viscosity solutions for the complex Monge-Amp\`ere equation. In particular it is shown here that the viscosity comparison principle holds for big cohomology classes whose polar and unbounded loci are equal