The intersection of two real forms in Hermitian symmetric spaces of compact type

Transcription

1 The intersection of two real forms in Hermitian symmetric spaces of compact type Makiko Sumi Tanaka (Tokyo University of Science) September 25, 2010 Tsinghua University 1

2 Joint with Hiroyuki Tasaki (University of Tsukuba) 1 Introduction 2 Preliminaries 3 Main results 4 Outline of proofs Contents 2

3 1 Introduction M = S 2 L 1, L 2 : great circles intersecting transversally = L 1 L 2 = {o, ō} (o and ō are antipodal) S 2 = CP 1 L 1, L 2 = RP 1 : totally geodesic R. Howard M = CP n L = RP n : totally geodesic = #(L g L) = n + 1 ( g I(M) s.t. g L intersects L transversally) x, y L g L are antipodal RP n is a real form of CP n. 3

4 2-number (B.-Y. Chen and T. Nagano) # 2 (RP n ) = n + 1 M. Takeuchi N : symmetric R-space = # 2 (N) = SB(N, Z 2 ) (=the sum of Z 2 -Betti numbers of N) Remark Every real form in a Hermitian symmetric space of compact type is a symmetric R-space. globally tight (Y.-G. Oh) M : Hermitian symmetric space L : Lagrangian submanifold def L : globally tight #(L g L) = SB(M, Z 2 ) for g I(M) s.t. g L intersects L transversally Remark : RP n is globally tight. 4

5 Problem 1 Does the intersection of two real forms of a Hermitian symmetric space of compact type consist of antipodal points if they intersect transversally? Moreover, does the number of such points coincide with the 2-number of the real form? Problem 2 Is every real form of any Hermitian symmetric space of compact type globally tight? 5

6 2 Preliminaries Definition M : connected Riemannian manifold M : Riemannian symmetric space def x M, s x : involutive isometry s.t. x is an isolated fixed point of s x s x is called the symmetry at x Definition M : connected Hermitian manifold M : Hermitian symmetric space def x M, s x : involutive holomorphic isometry s.t. x is an isolated fixed point of s x 6

7 If M is a Riemannian (resp. Hermitian) symmetric space, M = G/K with the identity component of the isometry group (resp. holomorphic isometry group) G and the isotropy subgroup K at some point in M. Definition M : Hermitian manifold L : real form of M def σ : M M: anti-holomorphic involutive isometry s.t. L = {x M σ(x) = x} Remark Real forms are connected. (M. Takeuchi) M : Hermitian symmetric space of compact type (i.e., M = G/K, G : compact, semisimple) = Every real form of M is a totally geodesic Lagrangian submanifold. 7

8 M : irreducible Hermitian symmetric space of compact type L : real form of M L M UI(n) = U(n)/SO(n) CI(n) = Sp(n)/U(n) Sp(n) = Sp(n) Sp(n)/ CI(2n) G r (R n+r ) = SO(n + r)/s(o(r) O(n)) G r (C n+r ) = SU(n + r)/s(u(r) U(n)) G r (H n+r ) = Sp(n + r)/sp(r) Sp(n) G 2r (C 2n+2r ) U(n) = U(n) U(n)/ G n (C 2n ) SO(n) = SO(n) SO(n)/ DIII(n) = SO(2n)/U(n) U II(n) = U(2n)/Sp(n) DIII(2n) S k S l /Z 2 G o 2 (Rk+l ) = SO(k + l)/so(2) SO(k + l 2) F II = F 4 /Spin(9) EIII = E 6 /T Spin(10) G 2 (H 4 )/Z 2 EIII T EIV = T E 6 /F 4 EV II = E 7 /T E 6 AII(4)/Z 2 EV II ( : the diagonal subgroup, T = U(1) ) M. Takeuchi, D.P.S. Leung 8

9 Definition (B.-Y. Chen - T. Nagano) M : Riemannian symmetric space S M : subset S : antipodal set def x, y S, s x y = y (s x : the symmetry at x) The 2-number of M # 2 (M) := sup{#s S M : antipodal set} If an antipodal set S satisfies #(S) = # 2 (M), S is called a great antipodal set. Remark # 2 (M) < Remark s x y = y c : closed geodesic s.t. x and y are antipodal on c Hence we can define the 2-number for any Riemannian manifold. 9

10 Examples # 2 (S n ) = 2 # 2 (T n ) = 2 n # 2 (KP n ) = n + 1 (K = R, C, H) e 1,..., e n+1 : o. n. b. of K n+1 = {Ke 1,..., Ke n+1 } : great antipodal set ( n ) # 2 (G r (K n )) =, the binomial coefficient (K = R, C, H) r e 1,..., e n : o. n. b. of K n = { e i1,..., e ir K 1 i 1 < < i r n} : great antipodal set # 2 (U(n)) = 2 n ±1... ±1 : great antipodal set 10

11 Definition M : compact Riemannian symmetric space, o M F (s o, M) = {x M s o (x) = x} = k j=0 M + j, where M + 0 = {o} Each connected component M + j is called a polar of M w.r.t. o. Examples (1) M = S n, o = (1, 0,..., 0) R n+1 F (s o, M) = {o} { o} (2) M = KP n (K = R, C, H) e 1,..., e n+1 : o.n.b. of K n+1, o = Ke 1 F (s o, M) = {Ke 1 } {1-dim subspaces in e 2,..., e n+1 K } = {o} KP n 1 11

12 (3) M = U(n), o = I : the identity matrix F (s o, M) = {X U(n) X 2 = I} = [I] [I 1 ] [I n 1 ] [I n ] = {I} G 1 (C n ) G n 1 (C n ) { I} [X] : conjugacy class of X I k = (the cardinality of 1 s is k and that of 1 s is n k) 12

13 Lemma 2.1 M : compact Riemannian symemtric space S : antipodal set of M, x S = S F (s x, M) Lemma 2.2 M : Hermitian symmetric space of compact type = M j + : Hermitian symmetric space of compact type Lemma 2.3 M : Hermitian symmetric space of compact type L : real form of M, o L = L M j + : real form of M j + if L M j + (M + j : polar of M w.r.t. o) 13

14 3 Main results Theorem 1 M: Hermitian symmetric space of compact type L 1, L 2 : real forms of M, intersect transversally = L 1 L 2 : antipodal set of L 1, L 2 Remark : We do not assume that L 1 and L 2 are congruent to each other. L 1 and L 2 are congruent if they are transformed to each other by some holomorphic isometry of M. 14

15 Theorem 2 M: Hermitian symmetric space of compact type L 1, L 2 : real forms of M, congruent, intersect transversally = L 1 L 2 : great antipodal set of L 1, L 2 i.e., #(L 1 L 2 ) = # 2 L 1 = # 2 L 2 Corollary Any real form of a Hermitian symmetric space of compact type is globally tight. 15

16 Theorem 3 M : irreducible Hermitian symmetric space of compact type L 1, L 2 : real forms of M, intersect transversally (1) (M, L 1, L 2 ) = (G 2m (C 4m ), G m (H 2m ), U(2m)) = #(L 1 L 2 ) = 2 m < # 2 L 1 < # 2 L 2 (2) (M, L 1, L 2 ) (G 2m (C 4m ), G m (H 2m ), U(2m)) = #(L 1 L 2 ) = min{# 2 L 1, # 2 L 2 } 16

17 4 Outline of proofs Theorem 1 M: Hermitian symmetric space of compact type L 1, L 2 : real forms of M, intersect transversally = L 1 L 2 : antipodal set of L 1, L 2 Proof of Theorem 1 Lemma (H. Tasaki) M : compact Kähler manifold with positive holomorphic sectional curvature L 1, L 2 : totally geodesic compact Lagrangian submanifolds in M = L 1 L 2 We can choose o L 1 L 2. We prove that o and p are antipodal for p L 1 L 2 {o} by using the properties of maximal tori. 17

18 Lemma M : compact Riemannian symmetric space, o M A, A 1 : maximal tori of M, o A A 1 S : fundamental cell of A, S = i S i A 1 A Exp S i = Exp S i A 1 A 18

19 Theorem 2 M: Hermitian symmetric space of compact type L 1, L 2 : real forms of M, congruent, intersect transversally = L 1 L 2 : great antipodal set of L 1, L 2 i.e., #(L 1 L 2 ) = # 2 L 1 = # 2 L 2 Proof of Theorem 2 We can choose o L 1 L 2. L 1, L 2 : congruent = # 2 L 1 = # 2 L 2 Let then F (s o, L i ) = F (s o, M) = r j=0 (L i M + j r j=0 M + j, ) (i = 1, 2) and L i M + j is a real form of M + j by Lemma

20 By Theorem 1 we have L 1 L 2 = and #(L 1 L 2 ) = Thus we obtain : r j=0 r j=0 {(L 1 M + j ) (L 2 M + j )} #{(L 1 M + j ) (L 2 M + j )}. ( ) j, #{(L 1 M + j ) (L 2 M + j )} = # 2(L 1 M + j ) = # 2(L 2 M + j ) = #(L 1 L 2 ) = # 2 L 1 = # 2 L 2 20

21 F (s o, M) = r M j + 1 j 1 =0 o j1 M + j 1, F (s oj1, M + j 1 ) = j 2 M + j 1,j 2.. o j1,...,j k M + j 1,...,j k, F (s oj1,...,j k, M + j 1,...,j k ) = j k+1 M + j 1,...,j k,j k+1 = dim M > dim M + j 1 > dim M + j 1,j 2 > > dim M + j 1,...,j l = 0 (i.e., M + j 1,...,j l = {a point}) L i M + j 1,...,j l (i = 1, 2) = #{(L 1 M + j 1,...,j l ) (L 2 M + j 1,...,j l )} = # 2 (L i M + j 1,...,j l ) = 1 (i = 1, 2) Now Theorem 2 is proved by induction due to ( ). 21

22 Theorem 3 M : irreducible Hermitian symmetric space of compact type L 1, L 2 : real forms of M, intersect transversally (1) (M, L 1, L 2 ) = (G 2m (C 4m ), G m (H 2m ), U(2m)) = #(L 1 L 2 ) = 2 m < # 2 L 1 < # 2 L 2 (2) (M, L 1, L 2 ) (G 2m (C 4m ), G m (H 2m ), U(2m)) = #(L 1 L 2 ) = min{# 2 L 1, # 2 L 2 } Proof of Theorem 3 We prove it case by case due to the classification of real forms of irreducible Hermitian symmetric spaces of compact type by D. P. S. Leung and M. Takeuchi. We may consider only the cases where L 1 and L 2 are not congruent. 22

23 M L 1 L 2 Q n (C) S k,n k S l,n l G C 2q (C2m+2q ) G H q (Hm+q ) G R 2q (R2m+2q ) G C n(c 2n ) U(n) G R n(r 2n ) G C 2m (C4m ) G H m (H2m ) U(2m) Sp(2m)/U(2m) Sp(m) U(2m)/O(2m) SO(4m)/U(2m) U(2m)/Sp(m) SO(2m) E 6 /T Spin(10) F 4 /Spin(9) G H 2 (H4 )/Z 2 E 7 /T E 6 T (E 6 /F 4 ) (SU(8)/Sp(4))/Z 2 23