| Co-Investigator(Kenkyū-buntansha) |
TOYONARI Toshitaka TMCAE Dep. of Gen. Education, Professor, 一般科, 教授 (20217582)
ONO Tomoaki TMCAE Dep. of Gen. Education, Professor, 一般科, 教授 (00224270)
中屋 秀樹 東京都立航空工業高等専門学校, 一般科, 助教授 (20271489)
TAKEI Kenji TMCAE Dep. of Gen. Education, Lecturer, 一般科, 講師 (90390426)
KADOWAKI Mituteru EHIME Univ., Dep. of Mechanical Engi., Asso. Professor, 工学部, 助教授 (70300548)
SUGIE Michio TMCAE Dep. of Gen. Education, Professor (90216309)
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| Budget Amount *help |
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Research Abstract |
First we consider smooth SU(p,q) actions on the complex projective (p+q-1)-space CP(p+q-1) whose restricted action to the maximal compact subgroup S(U(p)×U(q)) is standard. The first aim of this study is to construct such actions and to classify the actions up to the equivariant diffeomorphism classes.
Let (g,h_i) (i=1,2) be the pair of smooth functions g:RP(1)→R and h_i:U_i→R (U_1∪U_2= RP(1)) satisfying some differential equations and other three conditions. Then we can construct a smooth SU(p,q) action on CP(p+q-1) satisfying the previous condition. We already constructed such two pairs that the induced SU(p,q) actions are different. This time we found new three pairs. It is necessary to show that the actions induced from the pairs are same or not.
Next we consider smooth SL(m,C)×SL(n,C) actions on the complex projective (m+n-1)-space whose restricted action to the maximal compact subgroup SU(m)×SU(n) is standard. We also have studied such actions on the (m+n-1)-sphere.
Theorem 1 There exist infinitely many smooth SL(m,C)×SL(n,C) actions on CP(m+n-1) with three orbits, where two orbits are compact and one orbit is open.
Let Ψ be a smooth SL(m,C)×SL(n,C) action whose restricted action to SU(m)×SU(n) is standard. Then we can construct a R^2 actionΨ(c,d) on CP(1) for some real numbers c, d.
Theorem 2 Let Ψ, Ψ' be SL(m,C)×SL(n,C) actions whose restriction to SU(m)×SU(n) are standard. If Ψ(c,d)=Ψ'(c',d') and d-c=d'-c', thenΨ=Ψ'.
Theorem 3 Let α, β :R^2×CP(1)→CP(1) be the functions satisfying some three conditions, then we can construct an abstract SL(m,C)×SL(n,C) action on CP(m+n-1) whose restriction to SU(m)×SU(n) is standard.
In general, the action is not smooth.
In the near future, we will construct smooth SL(m,C)×SL(n,C) actions on CP(m+n-1) with k orbits (k>3) using this theorem.
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