( ) Kähler X ( ),. Floer -Oh- - [6]. X Fano *, X ( = (C ) N ) W : X C ( ) (X,W). X = P, W (y) =y + Q/y. Q P. Φ:X R N, Δ=Φ(X). u Int Δ, Lagrange L(u) =

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1 Floer Cohomologes of Non-torus Fbers of the Gelfand-Cetln System (X, ω) 2N. X N Φ=(ϕ,...,ϕ N ):X R N, Posson, Φ. Φ, Arnold-Louvlle Largange. Φ (u) = T N, ω Φ (u) =0.. Gelfand-Cetln, Gullemn-Sternberg [9] F = GL(n, C)/P. Gelfand-Cetln, Δ=Φ(F ) Gelfand-Cetln,, Δ Lagrange,., Lagrange Floer. Lagrange Floer, Lagrange Morse, Lagrange Hamltonan sotopy., Floer., Kähler X ( ). 97

2 ( ) Kähler X ( ),. Floer -Oh- - [6]. X Fano *, X ( = (C ) N ) W : X C ( ) (X,W). X = P, W (y) =y + Q/y. Q P. Φ:X R N, Δ=Φ(X). u Int Δ, Lagrange L(u) =Φ (u),. * 2 () L(u) u Int Δ PO(u, x) = H (L(u); R/2πZ) = Int Δ (R/2πZ) N v:(d 2, D 2 ) (X,L(u)) ( ) exp v ( ω hol x v( D 2 ) ) () D 2, W (y). hol x ( v( D 2 ) ), x H (L(u); R/2πZ) L(u) U() v( D 2 ) L(u). () PO, Lagrange L(u) b H (L(u); R/2πZ) (L(u),b), Floer. () X QH(X) Jacob Jac(PO) = C[y ±,...,y± N ]/( PO/ y ; =,...,N). (v), c (X) QH(X). * X Chern c (X) =c (TX), K X Rcc Kähler. *2,. ample. 98

3 [6] [7]., (), () X dm H (X; Q), Floer (L(u),b) dm H (X)., Gelfand-Cetln., Gvental [8], Batyrev, Cocan-Fontanne, Km, van Straten [] ( - - [0]). (), Floer Lagrange.,, dm H (F ). - -Xong [3] Retsch [] (C ) N,., Gelfand-Cetln., 3 Fl(3) C 4 2 Grassmann Gr(2, 4), Floer., Floer Lagrange dm H (F ). ( ). 2 Gelfand-Cetln u(n) n n Hermte, F = GL(n, C)/P λ =dag(λ,...,λ n ) O λ u(n). O λ λ,...,λ n Hermte. x O λ k =,...,n, x (k) x k k. x (k) Hermte, λ (k) (x) λ(k) 2 (x) λ(k) k (x) 99

4 . k =,...,n, n(n )/2 (λ (k) ) k n. λ λ 2 λ 3 λ n λ n λ (n ) λ (n ) 2 λ (n ) n λ (n 2) λ (n 2) n 2 (2) λ (). λ, F, (2) λ (k). λ (k) N =dm C F. Gelfand-Cetln λ (k). Φ=(λ (k) ),k : F R N 2. (Gullemn-Sternberg [9]). Φ F (Kostant-Krllov )., u Int Δ L(u) =Φ (u) Lagrange. Δ=Φ(F ) (2). Δ Gelfand-Cetln. 2.2 (Fl(3) ). λ,λ 2 > 0, Fl(3) λ =dag(λ, 0, λ 2 )., Gelfand-Cetln, 4 u 0 =(0, 0, 0) ( ). L 0 =Φ (u 0 ) L 0 = 0 0 z 0 0 z 2 u(3) z z 2 λ λ 2 z 2 + z 2 2 = λ λ 2 3 S 3 = SU(2) Lagrange. 00

5 λ () λ (2) λ (2) 2 Fl(3) Gelfand-Cetln. 2.3 (Gr(2, 4) ). λ>0, Gr(2, 4) λ =dag(λ, λ, λ, λ), Gelfand-Cetln Δ λ λ (3) 2 λ λ (2) λ (2) 2 λ () 4., λ () = λ (2) = λ (2) 2 = λ (2) 2 Δ. (t, t, t, t) L t =Φ (t, t, t, t) {( ) ti L t = 2 λ2 t 2 P λ2 t 2 P } u(4) ( t)i 2 P U(2) U(2) Lagrange. 3 Floer, -Oh- - [4] Floer. T, { } Λ 0 = a T λ a C, λ 0, lm λ = = 0

6 Novkov. Λ +,Λ. (X, ω) Lagrange L ( ), L, L H (L;Λ 0 ) A m k : H (L;Λ 0 ) k H (L;Λ 0 ), k =0,, 2,... ([4, Theorem A]). m : H (L) H (L), m 2 : H (L) H (L) H (L), m k (k 3). m m =0, m HF(L, L;Λ 0 )=Kerm / Im m L Floer. m m 0, Floer. b H (L;Λ + )( b H (L;Λ 0 )) A {m b k } k0. Floer m b m b (x) = k,l m k+l+ (b,...,b,x,b,...,b). }{{}}{{} k l. b Maurer-Cartan m k (b,...,b) 0 mod PD([L]) (3) k=0 m b m b =0., PD([L]) [L] Poncaré., m b HF((L, b), (L, b); Λ 0 )=Kerm b / Im m b (L, b) Floer. (3) weak boundng cochan, M weak (L). () PO, m k (b,...,b)=po(b) PD([L]) k=0 M weak (L). 02

7 , Cho-Oh [2, Secton 5], -Oh- - [5, Proposton 3.2, Theorem 3.4] Lagrange. Gelfand-Cetln Φ:F Δ Lagrange,.,. λ (k) θ (k) u =(u (k) ),k Int Δ L(u), b = (,k) I x (k) dθ (k) H (L(u); Λ 0 ) x =(x (k) ) (,k) I Λ N 0, H (L(u); Λ 0 ) Λ N 0. y (k) = e x(k) T u(k) Q j = T λ n j,, j =,...,r+,,. 3. ([0, Theorem 0.]). u Int Δ, H (L(u); Λ 0 ) M weak (L(u)). u Int Δ H (L(u); Λ 0 ) = Int Δ Λ N 0 PO(u, x) = (,k) I ( y (k+) y (k) ) + y(k) y (k+) +., λ (k+) = λ nj y (k+) = Q j Fl(3), PO = Q y + y Q 2 + Q 2 y 2 + y 2 Q 3 + y y 3 + y 3 y 2. dm H (Fl(3)) = 6., Floer (L(u),b) Grassmann Gr(2, 4), λ = λ 2 >λ 3 = λ 4, PO = Q y 2 + y 2 y + y y 3 + y 3 Q 3 + y 2 y 4 + y 4 y 3 03

8 , 4. Floer (L(u),b) 4., dmh (Gr(2, 4)) = 6, U(2)., Fl(3) Lagrange S 3 L 0. H (L 0 )=0, Floer m. 2.2, Fl(3) dag(λ, 0,λ 2 ) L 0 Fl(3) Novkov Λ 0 Floer HF(L 0,L 0 ;Λ 0 ) = Λ 0 /T mn{λ,λ 2 } Λ 0., Novkov Λ Floer : HF(L 0,L 0 ;Λ)=0. Fl(3), Λ Floer Lagrange Δ. Gr(2, 4) U(2) L t ( λ <t<λ) b H (L t ;Λ 0 /2π Z) = Λ 0 /2π Z, (L t,b) Floer HF((L t,b), (L t,b); Λ 0 ) = { H (L 0 ;Λ 0 ) t =0 b = ±π /2, (Λ 0 /T mn{λ t,λ+t} Λ 0 ) 2., Novkov Λ Floer. HF((L t,b), (L t,b); Λ) = { H (L 0 ;Λ) t =0 b = ±π /2, 0, Λ Floer (L, b), Gelfand-Cetln 6=dmH (Gr(2, 4)). 04

9 [] V. Batyrev, I. Cocan-Fontanne, B. Km, and D. van Straten, Mrror symmetry and torc degeneratons of partal flag manfolds, Acta Math. 84 (2000), no., 39. [2] C.-H. Cho and Y.-G. Oh, Floer cohomology and dsc nstantons of Lagrangan torus fbers n Fano torc manfolds, Asan J. Math. 0 (2006), [3] T, Eguch, K. Hor, and C-S. Xong, Gravtatonal quantum cohomology, Internat. J. Modern Phys. A 2 (997), no. 9, [4] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangan Intersecton Floer theory Anomaly and obstructons, Part I and Part II, AMS/IP Studes n Advanced Mathematcs, 46, [5] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangan Floer theory on compact torc manfolds I, Duke Math. J. 5 (200), no., [6] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangan Floer theory and mrror symmetry on compact torc manfolds, arxv: [7] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangan Floer theory on compact torc manfolds: survey, In Surveys n dfferental geometry. Vol. XVII, , Surv. Dffer. Geom., 7, Int. Press, Boston, MA (202). [8] A. Gvental, Statonary phase ntegrals, quantum Toda lattces, flag manfolds and the mrror conjecture, Topcs n sngularty theory, 03 5, Amer. Math. Soc. Transl. Ser. 2, 80, Amer. Math. Soc., Provdence, RI, 997. [9] V. Gullemn and S. Sternberg, The Gelfand-Cetln system and quantzaton of the complex flag manfolds, J. Funct. Annal. 52 (983), [0] T. Nshnou, Y. Nohara, and K. Ueda, Torc degeneratons of Gelfand-Cetln systems and potental functons, Adv. Math. 224, (200). [] K. Retsch, A mrror symmetrc constructon of qht (G/P ) (q), Adv. Math. 27 (2008), no. 6,