On the singularities of the pluricomplex Green's function

TL;DR

This paper demonstrates that on compact Kähler manifolds with boundary, the singularities of the pluricomplex Green's function with multiple poles can be precisely prescribed using local holomorphic functions, advancing understanding of complex Monge-Ampère equations.

Contribution

It introduces a method to prescribe singularities of the pluricomplex Green's function using blow-ups and recent estimates for degenerate complex Monge-Ampère equations.

Findings

Singularities can be prescribed as logarithms of sums of squares of holomorphic functions.

The proof combines blow-up techniques with $C^1$ estimates for degenerate Monge-Ampère equations.

The approach allows control over the Green's function's singularities at each pole.

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.