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j. differential geometry 72 (2006) 1-41
 

Summary: j. differential geometry
72 (2006) 1-41
ON THE ASYMPTOTIC EXPANSION OF BERGMAN
KERNEL
Xianzhe Dai, Kefeng Liu & Xiaonan Ma
Abstract
We study the asymptotic of the Bergman kernel of the spinc
Dirac operator on high tensor powers of a line bundle.
1. Introduction
The Bergman kernel in the context of several complex variables (i.e.,
for pseudoconvex domains) has long been an important subject (cf, for
example, [2]). Its analogue for complex projective manifolds is stud-
ied in [32], [29], [34], [14], [26], establishing the diagonal asymptotic
expansion for high powers of an ample line bundle. Moreover, the co-
efficients in the asymptotic expansion encode geometric information of
the underlying complex projective manifolds. This asymptotic expan-
sion plays a crucial role in the recent work of [22] where the existence
of K¨ahler metrics with constant scalar curvature is shown to be closely
related to Chow­Mumford stability.
Borthwick and Uribe [10], Shiffman and Zelditch [30] were the first

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara
Ma, Xiaonan - Institut de Mathématiques de Jussieu, Université Paris 7 - Denis Diderot

 

Collections: Mathematics