Construction of the topological toric theory
Project/Area Number 
13640087

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Osaka City University 
Principal Investigator 
MASUDA Mikiya Osaka City University, School of Science, Professor, 大学院・理学研究科, 教授 (00143371)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Period (FY) 
2001 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2002: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2001: ¥1,700,000 (Direct Cost: ¥1,700,000)

Keywords  toric variety / fan / convex polytope / combinatorics / topology / face ring / equivariant cohomology / elliptic genus / 組み合せ論 / トーラス作用 / 同変コホチロジー 
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 multifan. In particular, we found a neat formula describing the elliptic genus of a torus manifold in terms of the multifan 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 KobayashiOchiai 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 StanleyReisner 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 StanleyReisner 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 hvector 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.

Report
(3 results)
Research Products
(20 results)