Algebraicgeometrical and arithmetical study of a quotient space of a Riemannian symmetric space by an arithmetic group
Project/Area Number 
60540038

Research Category 
GrantinAid for General Scientific Research (C)

Allocation Type  Singleyear Grants 
Research Field 
代数学・幾何学

Research Institution  Kyoto University 
Principal Investigator 
FUJIKI Akira Kyoto University, Yoshida College, Associate Professor, 教養部, 助教授 (80027383)

CoInvestigator(Kenkyūbuntansha) 
KATO Shinichi Kyoto University, Yoshida College, Associate Professor, 教養部, 助教授 (90114438)
SAITO Hiroshi Kyoto University, Yoshida College, Associate Professor, 教養部, 助教授 (20025464)
AKIBA Tomoharu Kyoto University, Yoshida College, Professor, 教養部, 教授 (60027670)
TAKEUCHI Akira Kyoto University, Yoshida College, Professor, 教養部, 教授 (40026761)
FUJI'IE Tatsuo Kyoto University, Yoshida College, Professor, 教養部, 教授 (10026734)

Project Period (FY) 
1985 – 1986

Project Status 
Completed (Fiscal Year 1986)

Budget Amount *help 
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1986: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1985: ¥1,000,000 (Direct Cost: ¥1,000,000)

Keywords  Hermitian symmetric space / arithmetic subgroup / moduli space / 複素トーラス 
Research Abstract 
Quotinets of pseudohermitian symmetric sapces by certain arithmetic groups are realized often as a moduli space of various types of polarized compact Kahler manifolds, and through this fact their structures are clarified. From this point of view we have obtained the following results. 1. (1) For a general complex tori, we have classifieid the possible types of its rational endomorphism rings, and the corresponding pseudohermitian symmetric space. (2) We have obtained the complete classification of finite automorphism groups of complex tori of dimension two, which reflects the structure of the singularity of our quotient space. Especially, we have obtained explicit description of the moduli space of complex tori with fixed abstract group as automorphism group, as a quotient of a pseudohermitian space by arithmetic group. (3) We have found that the spaces in (2) are closely related with certain root lattices. 2. (1) As a general structure theorem for compact Kahler symplectic manifolds we have obtained an analogy of LefschetzHodge decomposition theorem, and some remarkable property of a natural 2nform on the second cohomology group. (2) As an application of (1) we have shown the smoothness of the local moduli space of sufh manifolds; this shows that their moduli space is essentially realized as an open subset of hermitian symmetric space of type IV. 3. In general, using the notion of extremal metrics due to Calabi, we have introduced the notion of stability in analogy with that in algebraic geometry, and obtained an entirely new formulation of the moduli theory. There seems much to be developped in the future in this theory.

Report
(1 results)
Research Products
(8 results)