On the singularities of the pluricomplex Green's function

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 $\log {\sum }_{j=1}^n\mathrm{\mid }{f}_j(z){\mathrm{\mid }}^2$ 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.