Project/Area Number  10440020 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
Geometry

Research Institution  Osaka University 
Principal Investigator 
FUJIKI Akira Graduate School of Science, Osaka University Professor, 大学院・理学研究科, 教授 (80027383)

CoInvestigator(Kenkyūbuntansha) 
GOTO Ryushi Graduate School of Science, Osaka University Lectwer, 大学院・理学研究科, 講師 (30252571)
WENG Lin 神戸大学, 理学部, 助教授 (60304002)
竹腰 見昭 大阪大学, 大学院・理学研究科, 助教授 (20188171)
MIYANISHI Masayoshi Graduate School of Science, Osaka University Professor, 大学院・理学研究科, 教授 (80025311)
SAKANE Yusuke Graduate School of Science, Osaka University Professor, 大学院・理学研究科, 教授 (00089872)
MABUCHI Toshiki Graduate School of Science, Osaka University Professor, 大学院・理学研究科, 教授 (80116102)
NAMIKAWA Yoshinori Graduate School of Science, Osaka University Associated Professor, 大学院・理学研究科, 助教授 (80228080)
榎 一郎 大阪大学, 大学院・理学研究科, 助教授 (20146806)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥5,800,000 (Direct Cost : ¥5,800,000)
Fiscal Year 1999 : ¥2,700,000 (Direct Cost : ¥2,700,000)
Fiscal Year 1998 : ¥3,100,000 (Direct Cost : ¥3,100,000)

Keywords  selfdual manifold / twistor space / group action / Joyce conjecture / complex manifold / toric surface / algebraic dimension / elliptic fiber space / 自己双対多様体 / twistor空間 / 群作用 / Joyce予想 / 複素多様体 / toric曲面 / 代数次元 / 楕円ファイバー空間 / Twistor空間 / ツイスター空間 / 自己双対計量 / ハイパーケーラー計量 
Research Abstract 
1. We have obtained an affimative answer to a conjugate of Joyce to the effect that a simply connected and complete selfdual manifold which admits a smooth action of a 2torus is diffeomorphic to a connected sum mCP (2) of m copies of complex projective plane and its selfdual structure is isomorphic to one of the examples constructed by Joyce himself in 1995. Our method was to use the twistor space associated to the given selfdual manifold. In fact, more generally, without simpleconnectivity assumption we have classified compact selfdual manifolds which admit an action of a two torus. As a byproduct of these investigation we could determine very precise structure as a complex manifold of the twistor space associated to Joyce selfdual metric. In particular we have shown that there exists a nice bimeromorphic model of the twistor space which is realized as a fiber space of torus surfaces and that this latter structure is completely determined by the invariant of the original smooth action of the torus on mCP (2). The twistor space is a Moishezon manifold and it is interesting to construct a natural birational projecitve model. 2. We have studied the deformation space of Joyce twistor space and have shown that its Kuranishi space is nonsingular. Furthermore, using such deformation we have given, for every integer m【greater than or equal】4 the first examples of selfdual structure of positive type on mCP (2) whose twistor space has algebraic dimension two. This result determines distribution of algebraic dimensions of twistor spaces associated to mCP (2). Furthermore, in the case of 4CP (2) we have given a detailed description of the twistor space as a general elliptic fiber space and then by taking its branched covering constructed a interesting family of compact complex manifolds whose canonical bundle is trivial.
