2002 Fiscal Year Final Research Report Summary
Construction of the topological toric theory
| Project/Area Number |
13640087
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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 | Osaka City University |
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Principal Investigator |
MASUDA Mikiya Osaka City University, School of Science, Professor, 大学院・理学研究科, 教授 (00143371)
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| Co-Investigator(Kenkyū-buntansha) |
HASHIMOTO Yoshitake Osaka City University, School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20271182)
HIBI Takayuki Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80181113)
TAKAKURA Tatsuru Chuo University, School of Science and Engineering, Lecturer, 理工学部, 講師 (30268974)
FURUSAWA Masaaki Osaka City University, School of Science, Professor, 大学院・理学研究科, 教授 (50294525)
KAWAYUCHI Akio Osaka City University, School of Science, Professor, 大学院・理学研究科, 教授 (00112524)
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| Project Period (FY) |
2001 – 2002
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| Keywords | toric variety / fan / convex polytope / combinatorics / topology / face ring / equivariant cohomology / elliptic genus |
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| Research Abstract |
We developed the theory of toric varieties from the topological viewpoint. In these several years I worked with Professor Akio Hattori and found that geometrical properies of a torus manifold can be described in terms of a combinatorial object called a multi-fan. In particular, we found a neat formula describing the elliptic genus of a torus manifold in terms of the multi-fan associated with the torus manifold, and obtained a vanishing theorem saying that the level N elliptic genus of a torus manifold vanishes if the 1st Chern class of the manifold is divisible by N. As a corollary of this vanishing theorem, we obtained a result that if the 1st Chern class of a complete toric variety M of complex dimension n is divisible by N, then N must be less than or equal to n+1, and in case N=n+l, M is isomorphic to the complex protective space. This is a toric version of the famous Kobayashi-Ochiai or Mori's theorem.
I invited Taras Panov from Moscow State University for a month and studied the equivariant cohomology of a torus manifold M and the cohomology of its orbit space. As a result, it turned out that when the cohomology ring of M is generated in degree two, the equivariant cohomology of M is a Stanley-Reisner ring and the orbit space of M has the same form as a convex polytope from a cohomological point of view. We also studied the case where M has vanishing odd degree cohomology. It turns out that this case is obtained by blowing down the previous case. Interestingly, the equivariant cohomology of M in this case provides a generalization of the Stanley-Reisner ring. The ring like this was already introduced by Stanley about ten years ago but we may think of our results as giving a geometrical meaning of the ring. Along this line, I proved a conjecture by Stanley about the h-vector of a Gorenstein* simplicial poset. The proof is purely algebraic but the idea stems from topology and this shows a close connection between combinatorics, commutative algebra and topology.
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