1999 Fiscal Year Final Research Report Summary
Topology of Sympletic Manifolds
| Project/Area Number |
10640093
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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 |
Geometry
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| Research Institution | Meiji University |
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Principal Investigator |
HATTORI Akio Meiji University, School of Science and Technology, Professor, 理工学部, 教授 (80011469)
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| Co-Investigator(Kenkyū-buntansha) |
MASUDA Mikiya Osaka City University, Faculty of Science,Professor, 理学部, 教授 (00143371)
AHARA Kazushi Meiji University, School of Science and Technology,Lecturer, 理工学部, 講師 (80247147)
SATO Atsushi Meiji University School of Science and Technology,Associate Professor, 理工学部, 助教授 (70178705)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Toric Variety / Torus Manifold / Fan / Multi-fan / Convex Polytope / Todd Genus / Riemann-Roch Number / Equivariant Cohomology |
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| Research Abstract |
The aim of the present research project was to study the relation between the structure of the multi-fan of a torus manifold and invariants of the manifold. In the course of research we succeeded in developing a general theory of combinatorics of multi-fans in a form suited to be applied to topological problems of torus manifolds. In this sense the original aim was attained. Main results will be stated in the following.
1. We defined a notion of TィイD2yィエD2 genus of a multi-fan, and showed that it coincides with the ordinary TィイD2yィエD2 genus of torus manifolds for multi-fans of the torus manifolds. We further showed an equality concerning the TィイD2yィエD2 genus similar to the one which holds between so-called h-vectors and f-vectors in combinatorics. Our formulation might be considered to give a new interpretation for the old quation.
2. We introduced the notion of multi-polytope in addition, and defined the Duistermaat-Heckman function and the winding number for multi-polytope. It was shown that the Duistermaat-Heckman function and the winding number determined each other. This generalizes a result known for multi-polytopes associated to torus manifolds. We also gave a generalization of multiplicity formula to the case of multi-fans.
3. Using the Duistermaat-Heckman function of a multi-fan a generalization of the Ehrhart polynomial is obtained. We showed the coefficient of the highest degree term coincided with the column of the multi-polytope and the constant term coincided with the Todd genus of the multi-polytope. The duality of the Ehrhart polynomial was also shown.
4. The Ehrhart polynomial of a convex polytope is closely related to the Riemann-Roch number of the corresponding ample line bundle over the relevant toric variety. In order to generalize this phenomenon to the case of multi-fans and multi-polytopes we defined the equivariant cohomology and Gysin homomorphism, and obtained a cohomological formula of Ehrhart polynomial.
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