Maximal subextensions of plurisubharmonic functions

Abstract

In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a subdomain of a compact Kahler manifold. We prove that a precise bound on the complex Monge-Ampere mass of the given function implies the existence of a subextension to a bigger regular subdomain or to the whole compact manifold. In some cases we show that the maximal subextension has a well defined complex Monge-Ampere measure and obtain precise estimates on this measure. Finally we give an example of a plurisubharmonic function with a well defined Monge-Ampere measure and the right bound on its Monge-Ampere mass on the unit ball in $\C^n$ for which the maximal subextension to the complex projective space $\mb P_n$ does not have a globally well defined complex Monge-Ampere measure.