Project/Area Number  10640093 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Geometry

Research Institution  Meiji University 
Principal Investigator 
HATTORI Akio Meiji University, School of Science and Technology, Professor, 理工学部, 教授 (80011469)

CoInvestigator(Kenkyūbuntansha) 
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)
藤田 宏 明治大学, 理工学部, 教授 (80011427)
SATO Atsushi Meiji University School of Science and Technology,Associate Professor, 理工学部, 助教授 (70178705)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1999 : ¥1,400,000 (Direct Cost : ¥1,400,000)
Fiscal Year 1998 : ¥1,600,000 (Direct Cost : ¥1,600,000)

Keywords  Toric Variety / Torus Manifold / Fan / Multifan / Convex Polytope / Todd Genus / RiemannRoch Number / Equivariant Cohomology / トーリック多様体 / 概複素多様体 / 特性類 / 種数 / 扇 / ベクトル束 
Research Abstract 
The aim of the present research project was to study the relation between the structure of the multifan of a torus manifold and invariants of the manifold. In the course of research we succeeded in developing a general theory of combinatorics of multifans 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 multifan, and showed that it coincides with the ordinary TィイD2yィエD2 genus of torus manifolds for multifans 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 socalled hvectors and fvectors in combinatorics. Our formulation might be considered to give a new interpretation for the old quation. 2. We introduced the notion of multipolytope in addition, and defined the DuistermaatHeckman function and the winding number for multipolytope. It was shown that the DuistermaatHeckman function and the winding number determined each other. This generalizes a result known for multipolytopes associated to torus manifolds. We also gave a generalization of multiplicity formula to the case of multifans. 3. Using the DuistermaatHeckman function of a multifan a generalization of the Ehrhart polynomial is obtained. We showed the coefficient of the highest degree term coincided with the column of the multipolytope and the constant term coincided with the Todd genus of the multipolytope. The duality of the Ehrhart polynomial was also shown. 4. The Ehrhart polynomial of a convex polytope is closely related to the RiemannRoch number of the corresponding ample line bundle over the relevant toric variety. In order to generalize this phenomenon to the case of multifans and multipolytopes we defined the equivariant cohomology and Gysin homomorphism, and obtained a cohomological formula of Ehrhart polynomial.
