Partially regular and cscK metrics

A Kahler metric g with integral Kahler form is said to be partially regular if the partial Bergman kernel associated to mg is a positive constant for all integer m sufficiently large. The aim of this paper is to prove that for all n >= 2 there exists an n-dimensional complex manifold equipped with strictly partially regular and cscK metric g. Further, for n >= 3, the (constant) scalar curvature of g can be chosen to be zero, positive or negative.