第61回トポロジーシンポジウム講演集 2014年7月 於 東北大学 Floer Cohomologies of Non-torus Fibers of the Gelfand-Cetlin System ݪ༤Ұ∗ ߳େֶ ڭҭֶ෦ 1 ং (X, ω) Λ 2N ࣍ݩγϯϓϨΫςΟοΫଟ༷ମͱ͢Δ. X ্ͷ N ݸͷؔͷ Φ = (ϕ1 , . . . , ϕN ) : X −→ RN ͕ؔతʹಠཱͰ, ʹ͍ޓPoisson Մ͋ͰΔͱ͖, Φ ΛશՄੵ͋ͰܥΔͱ͍͏. Φ ͷϑΝΠόʔ͕ίϯύΫτͳͱ͖, Arnold-Liouville ͷఆཧʹΑΓҰൠͷϑΝΠόʔ Largange τʔϥεͰ͋Δ. ͢ͳΘͪ Φ−1 (u) ∼ = TN, ω|Φ−1 (u) = 0 ͱͳΔ. ͜ͷయྫܕͷҰ͕ͭτʔϦοΫଟ༷ମ্ͷτʔϥε࡞༻ͷӡಈྔࣸ૾Ͱ͋Δ. දʹ͋Δ Gelfand-Cetlin ͱܥ, Guillemin-Sternberg [9] ʹΑΓಋೖ͞Εͨضଟ༷ମ F = GL(n, C)/P ্ͷશՄੵ͋ͰܥΔ. Gelfand-Cetlin ܥ, ͦͷ૾ Δ = Φ(F ) ͕ Gelfand-Cetlin ଟ໘ମͱΑΕΔತଟ໘ମʹͳΔͳͲ, τʔϦοΫଟ༷ମ্ͷӡಈྔࣸ૾ ͱΑ͘ࣅͨੑ࣭Λ͍࣋ͬͯΔ͕, Δ ͷڥք্ʹτʔϥεͰͳ͍ Lagrange ϑΝΠόʔΛ ࣋ͭͳͲ, τʔϦοΫଟ༷ମͷ߹ͱҟͳΔ໘͋Δ. ͜ͷߨԋͰ, ͦͷΑ͏ͳඇτʔ ϥε Lagrange ϑΝΠόʔͷ Floer ίϗϞϩδʔʹ͍ͭͯड़͍ͨ. Lagrange ෦ଟ༷ମͷ Floer ίϗϞϩδʔͱ, ͦͷ Lagrange ෦ଟ༷ମʹΛ ࣋ͭಓͷ্ۭؒͷ͋Δؔʹର͢Δ Morse ϗϞϩδʔͰ͋Γ, Lagrange ෦ଟ༷ମͷ Hamiltonian isotopy ʹؔ͢ΔෆมྔΛ༩͑Δ. ·ͨ, Floer ίϗϞϩδʔϛϥʔରশੑ ʹ͓͍ͯॏཁͳରͰ͋Δ. ϛϥʔରশੑͱ, K¨ ahler ଟ༷ମ X ͷγϯϓϨΫςΟο ∗ Պݚඅ (23740055) ͷॿΛड͚͍ͯΔ. 97 第61回トポロジーシンポジウム講演集 2014年7月 於 東北大学 ΫزԿ (͘͠ෳૉزԿ) ͱผͷ K¨ ahler ଟ༷ମ X ∨ ͷෳૉزԿ (γϯϓϨΫςΟοΫ زԿ) ͷ “Ձੑ” ͷ͜ͱͰ͋Γ, Ґ૬తͳ͔ؔΒݍతͳಉੑ·Ͱඇৗʹ͍༰Λ ؚΜͰ͍Δ. ͜͜ͰτʔϦοΫଟ༷ମͷ߹ͷ Floer ཧͱϛϥʔରশੑͷؔʹؔ͢Δਂ୩-Ohଠా-খ [6] ͷ݁ՌͷҰ෦Λ؆୯ʹࢥ͍ग़͓ͯ͘͠. X ͕τʔϦοΫ Fano ଟ༷ମ*1 ͷ ߹ʹ, ͦͷϛϥʔର X ∨ (∼ = (C∗ )N ) ͱͦͷ্ͷਖ਼ଇؔ W : X → C (εʔύʔϙ ςϯγϟϧͱݺΕΔ) ͷ (X ∨ , W ) ͱͳΔ. ྫ͑ X = P1 ͷ߹, εʔύʔϙςϯ γϟϧ W (y) = y + Q/y Ͱ༩͑ΒΕΔ. ͨͩ͠ Q P1 ͷγϯϓϨΫςΟοΫࣜܗͷ େ͖͞ʹରԠ͢ΔύϥϝʔλͰ͋Δ. Φ : X → RN Λτʔϥε࡞༻ͷӡಈྔࣸ૾ͱ͠, ӡ ಈྔଟ໘ମΛ Δ = Φ(X) ͱ͢Δ. ֤ u ∈ Int Δ ʹର͠, ͦͷ Lagrange τʔϥεϑΝ ΠόʔΛ L(u) = Φ−1 (u) ͱॻ͘͜ͱʹ͢Δͱ, ͕࣍Γཱͭ.*2 (i) L(u) ʹڥքΛ࣋ͭਖ਼ଇԁ൫Λ “͑Δ” ͜ͱʹΑΓఆٛ͞ΕΔ H 1 (L(u); R/2πZ) ∼ = Int Δ × (R/2πZ)N u∈Int Δ ্ͷϙςϯγϟϧؔ PO(u, x) = exp − v:(D2 ,∂D 2 )→(X,L(u)) D2 v ω holx v(∂D2 ) ∗ (1) , దͳมมͷͱͰεʔύʔϙςϯγϟϧ W (y) ͱҰக͢Δ. ͨͩ͠ holx v(∂D2 ) , x ∈ H 1 (L(u); R/2πZ) Λ L(u) ্ͷฏୱͳ U (1) ଓͱ͠ͳݟ ͨͱ͖ͷ v(∂D 2 ) ⊂ L(u) ʹԊͬͨϗϩϊϛʔͰ͋Δ. (ii) ϙ ς ϯ γ ϟ ϧ ؔ PO ͷ ྟ ք , Lagrange ϑ Ν Π ό ʔ L(u) ͱ b ∈ H 1 (L(u); R/2πZ) ͷ (L(u), b) Ͱ, Floer ί ϗ Ϟ ϩ δ ʔ ͕ ඇ ࣗ ໌ ͳ ͷʹରԠ͢Δ. (iii) X ͷྔࢠίϗϞϩδʔ QH(X) ϙςϯγϟϧؔͷ Jacobi Jac(PO) = ±1 ]/(∂PO/∂yi ; i = 1, . . . , N ) ͱಉ͋ͰܕΔ. C[y1±1 , . . . , yN (iv) ϙςϯγϟϧؔͷྟք, c1 (X) ∈ QH(X) ͷྔࢠΧοϓੵͷݻ༗ͱҰக ͢Δ. *1 *2 −1 ͕ ample ͳଟ༷ମͷ͜ͱ. ඍ X ͷୈ 1 Chern ྨ c1 (X) = c1 (T X) ͕ਖ਼, ͢ͳΘͪඪ४ଋ KX ahler ଟ༷ମ. زԿతʹ Ricci ͕ۂਖ਼ͷ K¨ ඇৗʹଟ͘ͷ͜ͱΛলུ͍ͯ͠ΔͷͰ, ਖ਼֬ͳओுͰͳ͍. 98 第61回トポロジーシンポジウム講演集 2014年7月 於 東北大学 ৄࡉ [6] [7] Λࢀর͞Ε͍ͨ. ಛʹϙςϯγϟϧؔͷྟք͕ͯ͢ඇୀԽͳΒ, (ii), (iii) ΑΓͦͷݸ X ͷίϗϞϩδʔ܈ͷ࣍ ݩdim H ∗ (X; Q) ʹҰக͠, ͕ͨͬ͠ ͯ Floer ίϗϞϩδʔ͕ඇࣗ໌ͳ (L(u), b) ͕ dim H ∗ (X) ݸଘࡏ͢Δ. ضଟ༷ମ্ͷ߹, τʔϦοΫୀԽΛ༻͍Δ͜ͱʹΑΓ Gelfand-Cetlin ܥͷτʔϥ εϑΝΠόʔͷϙςϯγϟϧؔΛ͢ࢉܭΔ͜ͱ͕Ͱ͖Δ. ͞Βʹ, ͦΕ͕ Givental [8], Batyrev, Ciocan-Fontanine, Kim, van Straten [1] ʹΑΓ༩͑ΒΕͨضଟ༷ମͷϛϥʔ ͷεʔύʔϙςϯγϟϧʹҰக͢Δ (ೲ-ݪ-২ా [10]). ͜ͷ߹্ͷ (ii) ͕Γཱ ͭͷͰ, ϙςϯγϟϧ͔ؔΒ Floer ίϗϞϩδʔ͕ඇࣗ໌ͳ Lagrange τʔϥεϑΝΠ όʔΛٻΊΔ͜ͱͰ͖Δ. ͔͠͠, τʔϦοΫଟ༷ମͷ߹ͱҟͳΓ, ͦͷҰൠ ʹ dim H ∗ (F ) ΑΓখ͍͞. ߐޱ-ງ-Xiong [3] Rietsch [11] ضଟ༷ମͷϛϥʔΛ తτʔϥε (C∗ )N ͷ෦ίϯύΫτԽͱͯ͠ߏ͠, ͦͷ্Ͱεʔύʔϙςϯγϟϧ ͕ਖ਼͍͠ͷྟքΛ࣋ͭ͜ͱΛ͍ࣔͯ͠Δ. ͜͜Ͱ৽ͨʹݱΕΔ “ແݶԕ” ͷྟք͕, ԿΒ͔ͷҙຯͰ Gelfand-Cetlin ଟ໘ମͷڥք্ʹ͋ΔඇτʔϥεϑΝΠόʔͱରԠͯ͠ ͍Δͱظ͢Δ͜ͱࣗવͳ͜ͱͩͱࢥΘΕΔ. ͜ͷߨԋͰ, 3 ࣍ݩඋضଟ༷ମ Fl(3) ͱ C4 ͷ 2 ࣍ݩ෦ۭؒͷͳ͢ Grassmann ଟ༷ମ Gr(2, 4) ͷ߹ʹ, ඇτʔϥεϑΝ Πόʔͷ Floer ίϗϞϩδʔͷࢉܭΛհ͍ͨ͠. ͦͷ݁Ռͱͯ͠, ͜ΕΒͷ߹ʹ Floer ίϗϞϩδʔ͕ফ͑ͳ͍ Lagrange ϑΝΠόʔͷ ͕ݸdim H ∗ (F ) ʹҰக͢Δ͜ ͱΛݟΔ. ͜Ε২ాҰੴࢯ (େࡕେֶ) ͱͷڞಉʹڀݚΑΔ. 2 Gelfand-Cetlin ܥ √ −1u(n) Λ n ߦ n ྻ Hermite ߦྻશମͷۭؒͱ͢Δͱ, ضଟ༷ମ F = GL(n, C)/P √ ͋Δର֯ߦྻ λ = diag (λ1 , . . . , λn ) ͷਵيಓ Oλ ⊂ −1u(n) ͱಉҰࢹͰ͖Δ. Oλ ݻ༗͕ λ1 , . . . , λn Ͱ͋ΔΑ͏ͳ Hermite ߦྻ͔ΒͳΔۭؒͰ͋Δ͜ͱʹҙ͢Δ. ֤ x ∈ Oλ ͱ k = 1, . . . , n − 1 ʹର͠, x(k) Λ x ͷࠨ্ͷ k × k ෦ߦྻͱ͢Δ. x(k) Hermite ߦྻ͔ͩΒ, ࣮ݻ༗ (k) (k) (k) λ1 (x) ≥ λ2 (x) ≥ · · · ≥ λk (x) 99 第61回トポロジーシンポジウム講演集 2014年7月 於 東北大学 Λ࣋ͭ. ͜ΕΛͯ͢ͷ k = 1, . . . , n − 1 ʹରͯ͠ߟ͑Δ͜ͱʹΑΓ, n(n − 1)/2 ݸͷؔ (k) ͷ (λi )1≤i≤k≤n−1 ͕ಘΒΕΔ. ͜ΕΒͷݻ༗ͨͪ (n−1) λn−1 ≥ ≥ ≥ ≥ ··· λ3 (n−1) (n−1) λ2 ≥ ≥ λn−1 ≥ λ1 λn ≥ λ2 ≥ λ1 (n−2) (n−2) λ1 λn−2 ≥ ≥ (2) · ·· · ·· ≥ ≥ (1) λ1 ͱ͍͏ؔΛຬͨ͢. λi ͨͪͷதʹ͍͠ͷ͕͋Δͱ͖, ͢ͳΘͪ F ͕උضଟ༷ମ (k) Ͱͳ͍߹, (2) ΑΓҰ෦ͷ λi (k) ͕ఆؔʹͳΔ. ఆͰͳ͍ λi ͷͪΐ͏Ͳ (k) N = dimC F ʹ͍͜͠ͱ͕͔Δ. Gelfand-Cetlin ܥఆͰͳ͍ λi ͨͪͷ Φ = (λi )i,k : F −→ RN (k) Ͱఆٛ͞ΕΔ. ໋ 2.1 (Guillemin-Sternberg [9]). ্ͷΑ͏ʹͯ͠ߏ͞Εͨ Φ ضଟ༷ମ F ্ͷ (Kostant-Kirillov ؔ͢ʹࣜܗΔ) શՄੵ͋ͰܥΔ. ·ͨ, ֤ u ∈ Int Δ ͷϑΝΠ όʔ L(u) = Φ−1 (u) Lagrange τʔϥεͰ͋Δ. ૾ Δ = Φ(F ) ෆࣜ (2) Ͱఆٛ͞ΕΔತଟ໘ମͱͳΔ. ͜ͷ Δ Λ Gelfand-Cetlin ଟ໘ମͱΑͿ. ྫ 2.2 (Fl(3) ͷ߹). λ1 , λ2 > 0 ΛͱΓ, Fl(3) Λ λ = diag(λ1 , 0, −λ2 ) ͷਵيಓͱ ಉҰࢹ͢Δ. ͜ͷͱ͖, Gelfand-Cetlin ܥ, 4 ຊͷล͕ू·͍ͬͯΔ u0 = (0, 0, 0) (ਤ 1 Ͱखલʹ͋Δ) ͰΒ͔Ͱͳ͍. ͜ͷ্ͷϑΝΠόʔ L0 = Φ−1 (u0 ) ⎧⎛ ⎫ ⎞ z1 ⎨ 0 0 ⎬ √ 2 2 ⎝ ⎠ 0 0 z2 L0 = ∈ −1u(3) |z1 | + |z2 | = λ1 λ2 ⎩ ⎭ z 1 z 2 λ1 − λ2 Ͱ༩͑ΒΕΔ 3 ࣍ٿݩ໘ S 3 ∼ = SU (2) ͱಉ૬ͳ Lagrange ෦ଟ༷ମͰ͋Δ. 100 第61回トポロジーシンポジウム講演集 2014年7月 於 東北大学 (1) λ1 (2) λ1 (2) λ2 ਤ1 Fl(3) ͷ Gelfand-Cetlin ଟ໘ମ. ྫ 2.3 (Gr(2, 4) ͷ߹). λ > 0 Λݻఆ͠, Gr(2, 4) Λ λ = diag(λ, λ, −λ, −λ) ͷਵي ಓͱಉҰࢹ͢Δͱ, Gelfand-Cetlin ଟ໘ମ Δ (3) −λ λ2 ≥ ≥ ≥ ≥ λ (2) (2) λ1 λ2 ≥ ≥ (1) λ1 (1) Ͱఆٛ͞ΕΔ 4 ࣍ݩತଟ໘ମͱͳΔ. ͜ͷ߹, τʔϥεͰͳ͍ϑΝΠόʔ λ1 (2) λ1 = (2) λ2 −1 Lt = Φ = (2) λ2 = Ͱఆ·Δ Δ ͷล্ʹݱΕΔ. ͜ͷล্ͷ (t, t, t, t) ্ͷϑΝΠόʔ (t, t, t, t) √ tI2 Lt = λ2 − t2 P ∗ √ λ2 − t2 P (−t)I2 ∈ √ −1u(4) P ∈ U (2) Ͱ༩͑ΒΕΔ U (2) ͱಉ૬ͳ Lagrange ෦ଟ༷ମͰ͋Δ. 3 ϙςϯγϟϧؔͱ Floer ίϗϞϩδʔ ݁ՌΛड़ΔͨΊʹ, ਂ୩-Oh-ଠా-খ [4] ʹΑΔ Floer ཧΛ؆୯ʹࢥ͍ग़͓ͯ͠ ͘. T Λࣜܗతͳύϥϝʔλͱͨ͠ͱ͖, Λ0 = ∞ i=1 ai T λi ai ∈ C, λi ≥ 0, lim λi = ∞ i→∞ 101 第61回トポロジーシンポジウム講演集 2014年7月 於 東北大学 Ͱఆٛ͞ΕΔॴہΛ Novikov ͏͍ͱ. ͦͷۃେΠσΞϧͱମΛͦΕͧΕ Λ+ , Λ ͱ ॻ͘͜ͱʹ͢Δ. γϯϓϨΫςΟοΫଟ༷ମ (X, ω) ͷ Lagrange ෦ଟ༷ମ L (ͱͦΕʹਵ͢Δ͍͘ ͔ͭͷσʔλ) ʹର͠, L ʹڥքΛ࣋ͭ֓ਖ਼ଇԁ൫Λ “্͑͛Δ” ͜ͱͰ, L ͷίϗϞϩ δʔ ܈H ∗ (L; Λ0 ) ্ʹ A∞ ߏ mk : H ∗ (L; Λ0 )⊗k −→ H ∗ (L; Λ0 ), k = 0, 1, 2, . . . ͕ఆ·Δ ([4, Theorem A]). m1 : H ∗ (L) → H ∗ (L) “ඍ” ͷΑ͏ͳͷͰ͋Γ, m2 : H ∗ (L) ⊗ H ∗ (L) → H ∗ (L) “ੵ”, mk (k ≥ 3) “ߴ࣍ͷੵ” Ͱ͋Δ. m1 ◦ m1 = 0 ͷͱ͖, m1 ͷఆΊΔίϗϞϩδʔ HF (L, L; Λ0 ) = Ker m1 / Im m1 Λ L ͷ Floer ίϗϞϩδʔͱ͍͏. Ұൠʹ m1 ◦ m1 = 0 Ͱ͋Γ, Floer ίϗϞϩδʔ ఆٛ͞ΕΔͱݶΒͳ͍. b ∈ H 1 (L; Λ+ ) (ྑ͍ঢ়Ͱگ b ∈ H 1 (L; Λ0 )) Λ༻͍ͯ A∞ ߏͷม{ ܗmbk }k≥0 Λ ߏ͢Δ͜ͱ͕Ͱ͖Δ. ྫ͑ม͞ܗΕͨ Floer ඍ mb1 mb1 (x) = k,l mk+l+1 (b, . . . , b, x, b, . . . , b). k l Ͱ༩͑ΒΕΔ. b ͕ Maurer-Cartan ํఔࣜ ∞ mk (b, . . . , b) ≡ 0 mod PD([L]) (3) k=0 Λຬ͍ͨͯ͠Δͱ͖ mb1 ◦ mb1 = 0 ͱͳΔ. ͨͩ͠, PD([L]) جຊྨ [L] ͷ Poincar´ e ରͰ͋Δ. ͜ͷͱ͖, mb1 ͷఆΊΔίϗϞϩδʔ HF ((L, b), (L, b); Λ0 ) = Ker mb1 / Im mb1 Λ (L, b) ͷ Floer ίϗϞϩδʔͱ͍͏. (3) ͷղΛ weak bounding cochain ͱΑͼ, weak (L) ͱॻ͘͜ͱʹ͢Δ. (1) Ͱ “ఆٛ” ͨ͠ϙςϯγϟϧؔ PO , ͦͷू߹Λ M ਖ਼֬ʹ ∞ mk (b, . . . , b) = PO(b) · PD([L]) k=0 weak (L) ্ͷؔͰ͋Δ. ʹΑͬͯఆٛ͞ΕΔ M 102 第61回トポロジーシンポジウム講演集 2014年7月 於 東北大学 τʔϦοΫଟ༷ମͷ߹ʹ, Cho-Oh [2, Section 15], ਂ୩-Oh-ଠా-খ [5, Propo- sition 3.2, Theorem 3.4] ʹΑΓ Lagrange τʔϥεيಓͷϙςϯγϟϧ͕ؔ͞ࢉܭ Ε͍ͯΔ. Gelfand-Cetlin ܥΦ : F → Δ ͷ Lagrange τʔϥεϑΝΠόʔʹରͯ͠, τʔϦοΫୀԽΛ༻͍Δ͜ͱͰϙςϯγϟϧؔΛ͢ࢉܭΔ͜ͱ͕Ͱ͖Δ. ͦͷ݁ՌΛड़ (k) ΔͨΊʹ, গ͠߸هͷ४උΛ͢Δ. ࡞༻ม λi u= (k) ͷରͰ͋Δ֯มΛ θi ͱ͠, (k) (ui )i,k ∈ Int Δ ্ͷϑΝΠόʔ L(u) ʹର͠, (k) (k) (k) xi dθi ∈ H 1 (L(u); Λ0 ) ←→ x = (xi )(i,k)∈I ∈ ΛN b= 0 (i,k)∈I ʹΑΓ H 1 (L(u); Λ0 ) ͱ ΛN 0 ΛಉҰࢹ͢Δ. (k) yi (k) (k) = exi T ui , Qj = T λnj , j = 1, . . . , r + 1, ͱ͓͘ͱ, ͕࣍Γཱͭ. ఆ ཧ 3.1 ([10, Theorem 10.1]). ֤ u ∈ Int Δ, ʹ ର ͠ H 1 (L(u); Λ0 ) ⊂ weak (L(u)) Ͱ͋Δ. ϙςϯγϟϧؔ M H 1 (L(u); Λ0 ) ∼ = Int Δ × ΛN 0 u∈Int Δ ্ͷؔͱͯ͠ PO(u, x) = (k) (i,k)∈I (k+1) Ͱ༩͑ΒΕΔ. ͨͩ͠, λi (k+1) yi (k) + yi yi (k+1) yi+1 (k+1) = λnj ͕ఆͷ߹ yi = Qj ͱ͢Δ. ྫ 3.2. 3 ࣍ضݩଟ༷ମ Fl(3) ͷ߹, ϙςϯγϟϧؔ PO = Q1 y1 Q2 y2 y1 y3 + + + + + y1 Q2 y2 Q3 y3 y2 Ͱ༩͑ΒΕΔ. ͜ͷྟքͷͪΐ͏Ͳ dim H ∗ (Fl(3)) = 6 Ͱ͋Δ. ͕ͨͬͯ͠, Floer ίϗϞϩδʔ͕ඇࣗ໌ͳ (L(u), b) 6 ͭଘࡏ͢Δ. ྫ 3.3. Grassmann ଟ༷ମ Gr(2, 4) ͷ߹, ͢ͳΘͪ λ1 = λ2 > λ3 = λ4 ͷͱ͖, ϙς ϯγϟϧؔ PO = Q1 y2 y1 y3 y2 y4 + + + + + y2 y1 y3 Q3 y4 y3 103 第61回トポロジーシンポジウム講演集 2014年7月 於 東北大学 ͱͳΓ, 4 ͭͷྟքΛ࣋ͭ. ͕ͨͬͯ͠ Floer ίϗϞϩδʔ͕ඇࣗ໌ͳ (L(u), b) 4 ͭ͋Δ. Ұํ, dim H ∗ (Gr(2, 4)) = 6 ͔ͩΒ, ͜ͷ߹ྟք͕ 2 ͭΓͳ͍. ͜ͷෆ͍ͯ͠ΔྟքʹରԠ͢Δͱظͨ͘͠ͳΔͷ͕ྫ 2.3 Ͱ ͨݟU (2) ϑΝΠ όʔͰ͋Δ. ͦΕΛݟΔલʹ, ·ͣ Fl(3) ͷ Lagrange S 3 ϑΝΠόʔ L0 Λߟ͑Δ. ͜ͷ߹ H 1 (L0 ) = 0 ͳͷͰ, Floer ඍ m1 ͷඇࣗ໌ͳมܗͳ͍. ྫ 2.2 ͱಉ༷, Fl(3) Λ diag(λ1 , 0, λ2 ) ͷਵيಓͱಉҰࢹ͢Δ. ఆཧ 3.4. L0 ⊂ Fl(3) ͷ Novikov Λ0 ্ͷ Floer ίϗϞϩδʔ HF (L0 , L0 ; Λ0 ) ∼ = Λ0 /T min{λ1 ,λ2 } Λ0 ͱͳΔ. ͕ͨͬͯ͠, Novikov ମ Λ ্ͷ Floer ίϗϞϩδʔࣗ໌Ͱ͋Δ: HF (L0 , L0 ; Λ) = 0. ಛʹ Fl(3) ͷ߹ʹ, Λ ͷ Floer ίϗϞϩδʔ͕ফ͑ͳ͍ Lagrange ϑΝΠόʔ Δ ͷ෦ʹݱΕΔτʔϥεͷΈͱͳΔ. ࣍ Gr(2, 4) ͷ U (2) ϑΝΠόʔͷ Lt (−λ < t < λ) ͷ߹Ͱ͋Δ. ͜͜Ͱྫ 2.3 ͱಉ͡ঢ়گΛߟ͑Δ. √ √ ఆཧ 3.5. b ∈ H 1 (Lt ; Λ0 /2π −1Z) ∼ = Λ0 /2π −1Z ʹର͠, (Lt , b) ͷ Floer ίϗϞϩ δʔ HF ((Lt , b), (Lt , b); Λ0 ) ∼ = H ∗ (L0 ; Λ0 ) (Λ0 /T min{λ−t,λ+t} Λ0 )2 √ t = 0 ͔ͭ b = ±π −1/2, ͦΕҎ֎ Ͱ༩͑ΒΕΔ. ͕ͨͬͯ͠, Novikov ମ Λ Λͱ͢Δ Floer ίϗϞϩδʔ HF ((Lt , b), (Lt , b); Λ) ∼ = H ∗ (L0 ; Λ) 0 √ t = 0 ͔ͭ b = ±π −1/2, ͦΕҎ֎ ͱͳΔ. ͭ·Γ, Λ ͷ Floer ίϗϞϩδʔ͕ඇࣗ໌ͳ (L, b) ͕, Gelfand-Cetlin ଟ໘ମͷ ͷϑΝΠόʔͱ߹Θͤͯͪΐ͏Ͳ 6 = dim H ∗ (Gr(2, 4)) ݸଘࡏ͍ͯ͠Δ. 104 第61回トポロジーシンポジウム講演集 2014年7月 於 東北大学 ࢀߟจݙ [1] V. Batyrev, I. Ciocan-Fontanine, B. Kim, and D. van Straten, Mirror symmetry and toric degenerations of partial flag manifolds, Acta Math. 184 (2000), no. 1, 1–39. [2] C.-H. Cho and Y.-G. Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006), 773–814. [3] T, Eguchi, K. Hori, and C-S. Xiong, Gravitational quantum cohomology, Internat. J. Modern Phys. A 12 (1997), no. 9, 1743.1782. [4] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Intersection Floer theory —Anomaly and obstructions—, Part I and Part II, AMS/IP Studies in Advanced Mathematics, 46, 2009. [5] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Floer theory on compact toric manifolds I, Duke Math. J. 151 (2010), no. 1, 23.174. [6] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Floer theory and mirror symmetry on compact toric manifolds, arXiv:1009.1648. [7] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Floer theory on compact toric manifolds: survey, In Surveys in differential geometry. Vol. XVII, 229–298, Surv. Differ. Geom., 17, Int. Press, Boston, MA (2012). [8] A. Givental, Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture, Topics in singularity theory, 103–115, Amer. Math. Soc. Transl. Ser. 2, 180, Amer. Math. Soc., Providence, RI, 1997. [9] V. Gullemin and S. Sternberg, The Gelfand-Cetlin system and quantization of the complex flag manifolds, J. Funct. Annal. 52 (1983), 106–128. [10] T. Nishinou, Y. Nohara, and K. Ueda, Toric degenerations of Gelfand-Cetlin systems and potential functions, Adv. Math. 224, 648–706 (2010). [11] K. Rietsch, A mirror symmetric construction of qHT∗ (G/P )(q) , Adv. Math. 217 (2008), no. 6, 2401–2442. 105
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