2005 Fiscal Year Final Research Report Summary
Complex analysis of L holomorphic functions on pseudoconvex domains of finite type
| Project/Area Number |
14340048
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | KYUSHU UNIVERSITY |
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Principal Investigator |
KAMIMOTO Joe Kyushu University, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (90301374)
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| Co-Investigator(Kenkyū-buntansha) |
IWASAKI Katsunori Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (00176538)
KAZAMA Hideaki Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10037252)
SATO Eiichi Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10112278)
TAKAGI Shunsuke Kyushu University, Faculty of Mathematics, Research Associate, 大学院・数理学研究院, 助手 (40380670)
KIMURA Hironobu Kumamoto University, Faculty of Mathematics, Professor, 大学院・自然科学研究科, 教授 (40161575)
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| Project Period (FY) |
2002 – 2005
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| Keywords | Bergman kernel / Singularity theory / Peak functions / Szegoe kernel / domains of finite type / toric variety / Asymptotic expansion / holomorphic line bundle |
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| Research Abstract |
We studied many kinds of objects in the function theory of several complex variables from the viewpoints of the theory of singularities. In particular, we are interested in the boundary behavior of the holomorphic functions which are square integrabel. Concretely the Bergman kernel and Szegoe kernel are very important integral kernel and they have many important information of the boundary behavior of holomotphic functions. As is very well known, the case of strictly pseudoconvex domains has many strong results about the Bergman kernel and Szegoe kernel. For example, the asymptotic expansion due to C.Fefferman reveals completely their boundary behaviors. We are interested in the weakly pseudoconvex domains case. The general case is very difficult to analyze and so we restrict ourselves to the objects in the case of finite type in the sense of D'Angelo. From the definition of finite type, the argument from algebraic geometry and singularity theory are valuable. We introduced the concepts of"Newton polyhedra"into the analysis of the Bergman kernel and showed that its singularity can be expressed in terms of the topological information of the Newton polyhedra. Moreover, we analyzed the construction of peak functions of any finite type domains.
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