On the singularities of the pluricomplex green's function

TLDR: In this article, it was shown that on a compact Kahler manifold with boundary, the singularities of the Green's function with multiple poles can be prescribed to be of the form Θ(log √ √ n|f_j(z)|^2$ at each pole, where n is the number of local holomorphic functions with the pole as their only common zero.

Abstract: It is shown that, on a compact Kahler manifold with boundary, the singularities of the pluricomplex Green's function with multiple poles can be prescribed to be of the form $\log\sum_{j=1}^n|f_j(z)|^2$ at each pole, where $f_j(z)$ are arbitrary local holomorphic functions with the pole as their only common zero. The proof is a combination of blow-ups and recent a priori estimates for the degenerate complex Monge-Ampere equation, and particularly the $C^1$ estimates away from a divisor.