1. No category

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 ‫ݸ‬ͷؔ਺ͷ૊
Φ = (ϕ1 , . . . , ϕN ) : X −→ RN
͕ؔ਺తʹಠཱͰ, ‫ ʹ͍ޓ‬Poisson Մ‫͋Ͱ׵‬Δͱ͖, Φ Λ‫׬‬શՄੵ෼‫͋Ͱܥ‬Δͱ͍͏. Φ
ͷϑΝΠόʔ͕ίϯύΫτͳͱ͖, Arnold-Liouville ͷఆཧʹΑΓҰൠͷϑΝΠόʔ͸
Largange τʔϥεͰ͋Δ. ͢ͳΘͪ
Φ−1 (u) ∼
= TN,
ω|Φ−1 (u) = 0
ͱͳΔ. ͜ͷయ‫ྫܕ‬ͷҰ͕ͭτʔϦοΫଟ༷ମ্ͷτʔϥε࡞༻ͷӡಈྔࣸ૾Ͱ͋Δ.
ද୊ʹ͋Δ Gelfand-Cetlin ‫ͱܥ‬͸, Guillemin-Sternberg [9] ʹΑΓಋೖ͞Εͨ‫ض‬ଟ༷ମ
F = GL(n, C)/P ্ͷ‫׬‬શՄੵ෼‫͋Ͱܥ‬Δ. Gelfand-Cetlin ‫ܥ‬͸, ͦͷ૾ Δ = Φ(F ) ͕
Gelfand-Cetlin ଟ໘ମͱΑ͹ΕΔತଟ໘ମʹͳΔͳͲ, τʔϦοΫଟ༷ମ্ͷӡಈྔࣸ૾
ͱΑ͘ࣅͨੑ࣭Λ͍࣋ͬͯΔ͕, Δ ͷ‫ڥ‬ք্ʹτʔϥεͰ͸ͳ͍ Lagrange ϑΝΠόʔΛ
࣋ͭͳͲ, τʔϦοΫଟ༷ମͷ৔߹ͱ͸ҟͳΔ໘΋͋Δ. ͜ͷߨԋͰ͸, ͦͷΑ͏ͳඇτʔ
ϥε Lagrange ϑΝΠόʔͷ Floer ίϗϞϩδʔʹ͍ͭͯड़΂͍ͨ.
Lagrange ෦෼ଟ༷ମͷ Floer ίϗϞϩδʔͱ͸, ͦͷ Lagrange ෦෼ଟ༷ମʹ୺఺Λ
࣋ͭಓͷ্ۭؒͷ͋Δؔ਺ʹର͢Δ Morse ϗϞϩδʔͰ͋Γ, Lagrange ෦෼ଟ༷ମͷ
Hamiltonian isotopy ʹؔ͢ΔෆมྔΛ༩͑Δ. ·ͨ, Floer ίϗϞϩδʔ͸ϛϥʔରশੑ
ʹ͓͍ͯ΋ॏཁͳର৅Ͱ͋Δ. ϛϥʔରশੑͱ͸, K¨
ahler ଟ༷ମ X ͷγϯϓϨΫςΟο
∗
Պ‫ݚ‬අ (23740055) ͷॿ੒Λड͚͍ͯΔ.
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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, Ґ૬తͳؔ܎͔Β‫ݍ‬࿦తͳಉ஋ੑ·Ͱඇৗʹ޿͍಺༰Λ
‫ؚ‬ΜͰ͍Δ.
͜͜Ͱ͸τʔϦοΫଟ༷ମͷ৔߹ͷ Floer ཧ࿦ͱϛϥʔରশੑͷؔ܎ʹؔ͢Δਂ୩-Ohଠా-খ໺ [6] ͷ݁ՌͷҰ෦Λ؆୯ʹࢥ͍ग़͓ͯ͘͠. X ͕τʔϦοΫ Fano ଟ༷ମ*1 ͷ
৔߹ʹ͸, ͦͷϛϥʔର͸ X ∨ (∼
= (C∗ )N ) ͱͦͷ্ͷਖ਼ଇؔ਺ W : X → C (εʔύʔϙ
ςϯγϟϧͱ‫ݺ‬͹ΕΔ) 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)
͸, ద౰ͳม਺ม‫׵‬ͷ΋ͱͰεʔύʔϙςϯγϟϧ 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) ͸ϙςϯγϟϧؔ਺ͷ Jacobi ‫ ؀‬Jac(PO) =
±1
]/(∂PO/∂yi ; i = 1, . . . , N ) ͱಉ‫͋Ͱܕ‬Δ.
C[y1±1 , . . . , yN
(iv) ϙςϯγϟϧؔ਺ͷྟք஋͸, c1 (X) ∈ QH(X) ͷྔࢠΧοϓੵͷ‫ݻ‬༗஋ͱҰக
͢Δ.
*1
*2
−1
͕ ample ͳଟ༷ମͷ͜ͱ. ඍ෼
X ͷୈ 1 Chern ྨ c1 (X) = c1 (T X) ͕ਖ਼, ͢ͳΘͪ൓ඪ४ଋ KX
ahler ଟ༷ମ.
‫ز‬Կతʹ͸ Ricci ‫͕཰ۂ‬ਖ਼ͷ K¨
ඇৗʹଟ͘ͷ͜ͱΛলུ͍ͯ͠ΔͷͰ, ਖ਼֬ͳओுͰ͸ͳ͍.
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. ಛʹϙςϯγϟϧؔ਺ͷྟք఺͕͢΂ͯඇୀԽͳΒ͹,
(ii), (iii) ΑΓͦͷ‫਺ݸ‬͸ X ͷίϗϞϩδʔ‫܈‬ͷ࣍‫ ݩ‬dim H ∗ (X; Q) ʹҰக͠, ͕ͨͬ͠
ͯ Floer ίϗϞϩδʔ͕ඇࣗ໌ͳ (L(u), b) ͕ dim H ∗ (X) ‫ݸ‬ଘࡏ͢Δ.
‫ض‬ଟ༷ମ্ͷ৔߹͸, τʔϦοΫୀԽΛ༻͍Δ͜ͱʹΑΓ Gelfand-Cetlin ‫ܥ‬ͷτʔϥ
εϑΝΠόʔͷϙςϯγϟϧؔ਺Λ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖Δ. ͞Βʹ, ͦΕ͕ Givental [8],
Batyrev, Ciocan-Fontanine, Kim, van Straten [1] ʹΑΓ༩͑ΒΕͨ‫ض‬ଟ༷ମͷϛϥʔ
ͷεʔύʔϙςϯγϟϧʹҰக͢Δ (੢ೲ-໺‫ݪ‬-২ా [10]). ͜ͷ৔߹΋্ͷ (ii) ͕੒Γཱ
ͭͷͰ, ϙςϯγϟϧؔ਺͔Β Floer ίϗϞϩδʔ͕ඇࣗ໌ͳ Lagrange τʔϥεϑΝΠ
όʔΛ‫ٻ‬ΊΔ͜ͱ΋Ͱ͖Δ. ͔͠͠, τʔϦοΫଟ༷ମͷ৔߹ͱ͸ҟͳΓ, ͦͷ਺͸Ұൠ
ʹ dim H ∗ (F ) ΑΓখ͍͞. ߐ‫ޱ‬-ງ-Xiong [3] ΍ Rietsch [11] ͸‫ض‬ଟ༷ମͷϛϥʔΛ୅
਺తτʔϥε (C∗ )N ͷ෦෼ίϯύΫτԽͱͯ͠ߏ੒͠, ͦͷ্Ͱεʔύʔϙςϯγϟϧ
͕ਖ਼͍͠਺ͷྟք఺Λ࣋ͭ͜ͱΛ͍ࣔͯ͠Δ. ͜͜Ͱ৽ͨʹ‫ݱ‬ΕΔ “ແ‫ݶ‬ԕ” ͷྟք఺͕,
ԿΒ͔ͷҙຯͰ Gelfand-Cetlin ଟ໘ମͷ‫ڥ‬ք্ʹ͋ΔඇτʔϥεϑΝΠόʔͱରԠͯ͠
͍Δͱ‫ظ‬଴͢Δ͜ͱ͸ࣗવͳ͜ͱͩͱࢥΘΕΔ. ͜ͷߨԋͰ͸, 3 ࣍‫׬ݩ‬උ‫ض‬ଟ༷ମ Fl(3)
ͱ C4 ಺ͷ 2 ࣍‫ݩ‬෦෼ۭؒͷͳ͢ Grassmann ଟ༷ମ Gr(2, 4) ͷ৔߹ʹ, ඇτʔϥεϑΝ
Πόʔͷ Floer ίϗϞϩδʔͷ‫ࢉܭ‬Λ঺հ͍ͨ͠. ͦͷ݁Ռͱͯ͠, ͜ΕΒͷ৔߹ʹ͸
Floer ίϗϞϩδʔ͕ফ͑ͳ͍ Lagrange ϑΝΠόʔͷ‫ ͕਺ݸ‬dim H ∗ (F ) ʹҰக͢Δ͜
ͱΛ‫ݟ‬Δ. ͜Ε͸২ాҰੴࢯ (େ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 ) ͷਵ൐‫ي‬ಓ Oλ ⊂ −1u(n) ͱಉҰࢹͰ͖Δ. Oλ
͸‫ݻ‬༗஋͕ λ1 , . . . , λn Ͱ͋ΔΑ͏ͳ Hermite ߦྻ͔ΒͳΔۭؒͰ͋Δ͜ͱʹ஫ҙ͢Δ.
֤ x ∈ Oλ ͱ k = 1, . . . , n − 1 ʹର͠, x(k) Λ x ͷࠨ্ͷ k × k ෦෼ߦྻͱ͢Δ. 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. ͜ΕΛ͢΂ͯͷ 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
ͱ͍͏ؔ܎Λຬͨ͢. λi ͨͪͷதʹ౳͍͠΋ͷ͕͋Δͱ͖, ͢ͳΘͪ F ͕‫׬‬උ‫ض‬ଟ༷ମ
(k)
Ͱ͸ͳ͍৔߹, (2) ΑΓҰ෦ͷ λi
(k)
͕ఆ਺ؔ਺ʹͳΔ. ఆ਺Ͱͳ͍ λi
ͷ਺͸ͪΐ͏Ͳ
(k)
N = dimC F ʹ౳͍͜͠ͱ͕෼͔Δ. Gelfand-Cetlin ‫ܥ‬͸ఆ਺Ͱͳ͍ λi
ͨͪͷ૊
Φ = (λi )i,k : F −→ RN
(k)
Ͱఆٛ͞ΕΔ.
໋୊ 2.1 (Guillemin-Sternberg [9]). ্ͷΑ͏ʹͯ͠ߏ੒͞Εͨ Φ ͸‫ض‬ଟ༷ମ F ্ͷ
(Kostant-Kirillov ‫ؔ͢ʹࣜܗ‬Δ) ‫׬‬શՄੵ෼‫͋Ͱܥ‬Δ. ·ͨ, ֤಺఺ u ∈ Int Δ ͷϑΝΠ
όʔ L(u) = Φ−1 (u) ͸ Lagrange τʔϥεͰ͋Δ.
૾ Δ = Φ(F ) ͸ෆ౳ࣜ (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 ) ͷਵ൐‫ي‬ಓͱ
ಉҰࢹ͢Δ. ͜ͷͱ͖, Gelfand-Cetlin ‫ܥ‬͸, 4 ຊͷล͕ू·͍ͬͯΔ௖఺ u0 = (0, 0, 0)
(ਤ 1 Ͱखલʹ͋Δ௖఺) Ͱ‫׈‬Β͔Ͱͳ͍. ͜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 ෦෼ଟ༷ମͰ͋Δ.
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) ͷਵ൐‫ي‬
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 ෦෼ଟ༷ମͰ͋Δ.
3 ϙςϯγϟϧؔ਺ͱ Floer ίϗϞϩδʔ
݁ՌΛड़΂ΔͨΊʹ, ਂ୩-Oh-ଠా-খ໺ [4] ʹΑΔ Floer ཧ࿦Λ؆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
Ͱఆٛ͞ΕΔ‫؀ॴہ‬Λ Novikov ‫͏͍ͱ؀‬. ͦͷ‫ۃ‬େΠσΞϧͱ঎ମΛͦΕͧΕ Λ+ , Λ ͱ
ॻ͘͜ͱʹ͢Δ.
γϯϓϨΫςΟοΫଟ༷ମ (X, ω) ͷ Lagrange ෦෼ଟ༷ମ L (ͱͦΕʹ෇ਵ͢Δ͍͘
͔ͭ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෼” ͷΑ͏ͳ΋ͷͰ͋Γ,
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͸
ఆٛ͞ΕΔͱ͸‫ݶ‬Βͳ͍.
b ∈ H 1 (L; Λ+ ) (ྑ͍ঢ়‫Ͱگ‬͸ b ∈ H 1 (L; Λ0 )) Λ༻͍ͯ A∞ ߏ଄ͷม‫{ ܗ‬mbk }k≥0 Λ
ߏ੒͢Δ͜ͱ͕Ͱ͖Δ. ྫ͑͹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) ্ͷؔ਺Ͱ͋Δ.
ʹΑͬͯఆٛ͞ΕΔ M
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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 τʔϥε‫ي‬ಓͷϙςϯγϟϧؔ਺͕‫͞ࢉܭ‬
Ε͍ͯΔ. Gelfand-Cetlin ‫ ܥ‬Φ : F → Δ ͷ Lagrange τʔϥεϑΝΠόʔʹରͯ͠͸,
τʔϦοΫୀԽΛ༻͍Δ͜ͱͰϙςϯγϟϧؔ਺Λ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖Δ. ͦͷ݁ՌΛड़
(k)
΂ΔͨΊʹ, গ͠‫߸ه‬ͷ४උΛ͢Δ. ࡞༻ม਺ λi
u=
(k)
ͷ૒ରͰ͋Δ֯ม਺Λ 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
্ͷؔ਺ͱͯ͠
PO(u, x) =
(k)
(i,k)∈I
(k+1)
Ͱ༩͑ΒΕΔ. ͨͩ͠, λi
(k+1)
yi
(k)
+
yi
yi
(k+1)
yi+1
(k+1)
= λnj ͕ఆ਺ͷ৔߹͸ 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. ͜ͷྟք఺ͷ਺͸ͪΐ͏Ͳ 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
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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 ͭͷྟք఺Λ࣋ͭ. ͕ͨͬͯ͠ Floer ίϗϞϩδʔ͕ඇࣗ໌ͳ (L(u), b) ΋ 4
ͭ͋Δ. Ұํ, dim H ∗ (Gr(2, 4)) = 6 ͔ͩΒ, ͜ͷ৔߹͸ྟք఺͕ 2 ͭ଍Γͳ͍.
͜ͷෆ଍͍ͯ͠Δྟք఺ʹରԠ͢Δͱ‫ظ‬଴ͨ͘͠ͳΔͷ͕ྫ 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 ͷඇࣗ໌ͳม‫ܗ‬͸ͳ͍. ྫ 2.2 ͱಉ༷, Fl(3) Λ
diag(λ1 , 0, λ2 ) ͷਵ൐‫ي‬ಓͱಉҰࢹ͢Δ.
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 ίϗϞϩδʔ͸ࣗ໌Ͱ͋Δ:
HF (L0 , L0 ; Λ) = 0.
ಛʹ Fl(3) ͷ৔߹ʹ͸, Λ ܎਺ͷ Floer ίϗϞϩδʔ͕ফ͑ͳ͍ Lagrange ϑΝΠόʔ
͸ Δ ͷ಺෦ʹ‫ݱ‬ΕΔτʔϥεͷΈͱͳΔ.
࣍͸ Gr(2, 4) ಺ͷ U (2) ϑΝΠόʔͷ଒ Lt (−λ < t < λ) ͷ৔߹Ͱ͋Δ. ͜͜Ͱ͸ྫ
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 Λ܎਺ͱ͢Δ 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.
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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
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