*† Dep. of Mathematical Sciences, KAIST
1 ( )
( ) ([30]
1 1 [5, 17, 18] ( 2
)
ϕ : G × M → M (1) ϕ(g,ϕ(h,x)) = ϕ(gh,x); (2) ϕ(e,x) = x
2
1

3 (M,G)
1
M/G [ x ] G(x) T C G x T (G z ) M × G N G- (G- ) Theorem 2.1. x ∈ M G(x) ≡ G/K X G- .. X ⇒ G × K D x , dim M − dim G(x) = k x )               K − D x ( σ : K → O(D z ) ~ O(k z ) Theorem 2.1 X M G- k: n Φ(g) = φ(g, ·) Φ : G → Diff( M ) *2 (2.1) G : (2.2) ( ) X N 1 2 3.1 (M,G) 0 n
G- 0 x ∈ M : dim G( x ) = n. G
M 1
1 3.2 (M,G) M G- n :
(Q ≠ [0, 1]
x ∈ M G 1
1 dim G(x) = n − 1.
1
(1) Q = R, ; (2) Q = R+, ; (3) Q = S¹,       ; (4) Q = [0, 1], . M G- (Q = R, R+, S1) 3.2 3.2.1 Q = R元:
K ⊂ G
G/K
G/K- R spherical varietyし*3 C ×4 ) ([39] 0 ) G ([6] Q = R 3.1 0 Q = R+ 3.2.2
G(x) ≡ G/K y ∈ X
)
M ≥ G × K V, (3.2)
Q = R+ G H ⊂ K K/H ≥ Sℓ− 1 (ℓ = n − dim G − dim K) ） ( ) ([16] 1950 Borel[4], Montgomery-Samelson[31], Poncet[35] 3.2.3
Q = S1
G/K
4
([1, 3, 5, 11, 13, 19, 20, 21, 23, 24, 34, 37, 40] Uchida[37] ( (Hattori-Masuda [15, 26]) ） 3.3 Wang-Uchida Wang[40] : : S1 1 ) 3.3.2 Hattori-Masuda[15] ) 2n Q = [0, 1]
MT M T = { x ∈ M | T(x) = x } T M MT ( $2n , Tn ) (CP(n), T n ) 2n (t 1 , . . . , t n ) ∈ T n (z1,...,zn,r) E S²n ⊂ Cn ⊕ R ( z 1 , .., zn,r) → (t 1 z 1, ..., tnzn,r). 2 , t n ) ∈ T n (t 1 , n :
n n + 1 T n PU(n + 1) SU (n + 1) ) : C" = R2n Cn+1 C r2 ( U(n) SO(2n) PU(n + 1) = SU(n + 1)/C( C 1 *5
»6 S 1 n n 2 omniorientation ) ([26] ( M , T ) T- ( +6. *5
M
? 3.3.3 Wang-Uchida Problem 1 Wang-Uchida Problem 1 STEP 1. Wang-Uchida
(
Lemma 3.1. G K i (i = 1, 2) G ≈ Π j =1 PU(ℓ j + 1) × Π h =1 SO(2mh + 1) × G j , K j ≈ Π j =1 P(U(ℓ j ) × U(1)) × Π h =1 SO(2mh ) × G j . G/K i ≈ Π j =1 CP( l j ) × Π h =1 S 2 m h . }} Lemma 3.1
G- STEP 3.
N G ( K )/K
(Q = S 1 STEP 3 STEP 4 T X 1 , X 2 7 T ([37] ) Lemma 3.2 (Uchida's criterion). X 1 U 7 X 2 X 1 U 7 X 2 (1)  τ G-diffeotopic; (2) τ−1τ′ T
STEP 4
Theorem 3.3 ([23]). G-
7 G- ( N, H ) :
P(C k 1 ⊕ C k 2 ) dimc = k 1 + k 2 − 1 Theorem 3.3
S(V ⊕ W) ⊂ V ⊕ W
( Wang-Uchida
1 4 Theorem 3.3 4.1
? Theorem 3.3 1 1 [29] Problem 2 (cohomological rigidity problem). ??
4.2
k = 1 ) 4.2 k:
G- k: k
3.3 *8) Davis-Januszkiewicz[9]
4.2.2 Theorem 3.3
T- G- Demazure
*9 Masuda[28] 1 Wiemeler[41] admissible 5-tuple Theorem 3.3 SU, SO (A, B, D ) ●10 (toric variety A GKM
G/H GKM [1] A.V. Alekseevskii, D.V. Alekseevskii, G-manifolds with one-dimensional orbit space , Adv. in Soviet Math., 8 (1992), 1–31. [2] I.V. Arzhantsev, S.A. Gaifullin, Homogeneous toric varieties, J. Lie Theory, 20 (2010), 283-293. [3] T. Asoh, Compact transformation groups on Z 2 -cohomology spheres with orbits of codimension 1, Hirosima Math. J., 11 (1981), 571–616. [4] A. Borel, Le plan projectif de actaves et les spheres comme espaces homogenes , C.R. Acad. Sci., Paris, 230 (1950), 1378-1381. [5] G.E. Bredon, Introduction to compact transformation groups, Academic Press, 1972. [6] M. Brion, Spherical varieties, the lecture note written by R. Devyatov, D. Fratila, V. Tsanov, http://www- fourier.ujf-grenoble.fr/~mbrion/SphericalVar.pdf, (2009). [7] S. Choi, S. Kuroki, Topological classification of torus manifolds which have codimension one extended actions, arXiv:0906.1335; OCAMI preprint series 09-9, (2009). [8] M. Davis, Smooth G-manifolds as collections of fiber bundles , Pacific J. of Math., 77 (1978), 315–363. [9] M. Davis, T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus action, Duke. Math. J., 62 (1991), 417–45. toric variety autmorphism group Autmorphism group (C*) n 2010 5 Fudan University Masuda
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