On the singularities of the pluricomplex Green's function

2012/09/11 by D. H. Phong, Phong, D. H., J. Sturm +1
Mathematics · #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1209.2198

openalex publication_date 2012/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∑_j=1ⁿ|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 C¹ estimates away from a divisor.

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