2000 Fiscal Year Final Research Report Summary
Topological research of the theory of toric varieties
| Project/Area Number |
11640091
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Osaka City University |
|---|
Principal Investigator |
MASUDA Mikiya Osaka City University, School of Science, Professor, 医学部, 教授 (00143371)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
HASHIMOTO Yoshitake Osaka City University, School of Science, Associate Professor, 理学部, 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)
|
|---|
| Project Period (FY) |
1999 – 2000
|
| Keywords | toric variety / fan / convex polytope / combinatorics / topology / equivariant cohomology / Ehrhart polynomial |
|---|
| Research Abstract |
We developed the theory of toric varieties from topological viewpoint. The theory of toric varieties says that there is a one-to-one correspondence between "toric varieties" (an object in algebraic geometry) and "fans" (an object in combinatorics). In our project, we studied "torus manifolds" or "torus orbifolds" which are topological counterparts to toric varieties and a wider object than that of toric varieties, and constructed a correspondence from those extended objects to an extended combinatorial object called "multi-fans". One of the fundamental problems in our correspondence is to characterize geometrically obtained multi-fans, and we completely characterized the multi-fans obtained form torus orbifolds. Moreover, we described signatures and T_y-genera of torus manifolds in terms of multi-fans. There is another fundamental correspondence given by moment maps. We introduced a notion of multi-polytopes, which appear as images of moment maps, and generalized Ehrhart polynomials and Khovanskii-Pukhlikov formula for convex polytopes to multi-polytopes.
|
