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Advances in AnalysisThe Legacy of Elias M. Stein$
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Charles Fefferman, Alexandru D. Ionescu, D.H. Phong, and Stephen Wainger

Print publication date: 2014

Print ISBN-13: 9780691159416

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159416.001.0001

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On the Singularities of the Pluricomplex Green’s Function

On the Singularities of the Pluricomplex Green’s Function

Chapter:
(p.419) Chapter Sixteen On the Singularities of the Pluricomplex Green’s Function
Source:
Advances in Analysis
Author(s):

D. H. Phong

Jacob Sturm

, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159416.003.0016

This chapter seeks to establish the existence of pluricomplex Green's functions with singularities at certain multi poles, given by arbitrary local analytic functions. The Green's function plays a central role in the study of functions of one complex variable or of two real variables. This chapter also attempts to develop a geometric/analytic approach to Monge-Ampère equations with measures on the right-hand side, where the singularities of the solution arise from blow-up constructions. Since blow-ups typically lead to degenerate Kähler forms, an essential tool in this chapter's approach is the recent existence theorems for the Dirichlet problem for complex Monge-Ampère equations with degenerate background form.

Keywords:   pluricomplex Green's functions, geometric approach, Monge-Ampère equations, singularities, blow-ups, Kähler forms

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