1998 Fiscal Year Final Research Report Summary
Studies on Hodge Theory and Hypergeometric Functions
| Project/Area Number |
09640010
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | THE UNIVERASITY OF TOKYO |
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Principal Investigator |
KAWAMATA Yujiro University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (90126037)
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| Co-Investigator(Kenkyū-buntansha) |
SAITO Takeshi University of Tokyo, Graduate School of Mathematical Sciences, Associate Profess, 大学院・数理科学研究科, 助教授 (70201506)
KATSURA Toshiyuki University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40108444)
ODA Takayuki University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (10109415)
TERASOMA Tomohide University of Tokyo, Graduate School of Mathematical Sciences, Associate Profess, 大学院・数理科学研究科, 助教授 (50192654)
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| Project Period (FY) |
1997 – 1998
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| Keywords | alsebraic variety / pluricanonical form / Kodaira dimension / logarithmic form / multiplier ideal / canonical singularity / deformation / plurigenus |
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| Research Abstract |
In the course of the investigation of the structure of algebraic varieties, it is often very helpful to look at the pluricanonical forms on the given varieties. For example, the Kodaira dimension of a variety is the order of growth of the m-genus, the dimension of the vector space of m-canonical forms, as a function on the integer m. This is a fundamental invariant for the birational classification of algebraic varieties. We investigated the problem of comparing log pluricanonical forms on a given variety with pluricanonical forms on its subvariety.
Let X be a smooth algebraic variety and Y a smooth divisor. For example, Y is a compact algebraic manifold and X is the total space of its deformation family. We considered the problem of extending pluricanonical forms on Y to log pluricanonical forms on X.We defined sequences of multiplier ideal sheaves on X and Y, and the extension problem is reduced to the problem of the inclusion relationships among these ideals. We proved an extension theorem under some conditions. As applications. we proved that any deformations of canonical singularities are canonical, and that the plurigenus is constant on any deformation family of varieties of general type with only canonical singularities.
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