Invariant theory of the Bergman kernel and index theorems.
Project/Area Number  10640168 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Osaka University 
Principal Investigator 
HIRACHI Kengo Graduate School of Science, Osaka Univ. Lecturer, 大学院・理学研究科, 講師 (60218790)

CoInvestigator(Kenkyūbuntansha) 
OHTSU Yukio Graduate School of Science, Osaka Univ. Lecturer, 大学院・理学研究科, 講師 (80233170)
TAKEGOSHI Kensho Graduate School of Science, Osaka Univ. Associate Prof., 大学院・理学研究科, 助教授 (20188171)
KOMATSU Gen Graduate School of Science, Osaka Univ. Associate Prof., 大学院・理学研究科, 助教授 (60108446)

Project Period (FY) 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,200,000 (Direct Cost : ¥3,200,000)
Fiscal Year 1999 : ¥1,200,000 (Direct Cost : ¥1,200,000)
Fiscal Year 1998 : ¥2,000,000 (Direct Cost : ¥2,000,000)

Keywords  Bevgman Kevuel / Parabolic Tnvaviant theory / CR invariant / index theorem / セゲー核 / グラウエルト柱状領域 
Research Abstract 
This project is an attempt to give relations between the local and the global biholomorphic invariants of strictly pseudoconvex domains by using the Bergman kernel. We have obtained the following two results : (1) A relation between the Bergman kernel of Grauert tube and Hilbert polynomial. For an ample line bundle L on a projective manifold, the dimension of the space of the holomorphic sections of mthe power of L is given by a polynomial P(m) in m. P(m) is called Hilbert polynomial and is a global in variant of L. We gave an explicit relation between P(m) and the asymptotic expansion of the Bergman kernel for the Grauert tube in the dual bundle of L. The relation is given by Laplece transform, and it shows that the characteristic class of L appears in the asymptotic expansion of the Bergman kernel. (2) Analytic continuation of SobolevBergman kernels with respect to the Sobolev order. We generalized the invariant theory of the Bergman kernel to a class of SobolevBergman kernels. We first construct SobolevBergman kernels in such a way that they satisfy biholomorphic transformation law and that their boundary asymptotics are given by local biholomorphic invariants. We then showed that the kernels can be analytically continued to a meromorphic function on the complex plain with respect to the Sobolev order. We further proved that the universal constants in the asymptotic expansion of SobolevBergman kernel are polynomials in the Sobolev order. This enables us to compute the analytic continuation of the kernels explicitly.

Report
(3results)
Research Products
(9results)