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Transcript of Higgs #Å #¯ Kobayashi-HitchinHiggs #Å #¯Kobayashi-Hitchin; +#¬#¥# # w O ^ ë P E = J /#«;...
Higgs Kobayashi-Hitchin
,. , (Kobayashi-Hitchin ) .
1960 Narasimhan Seshadri Riemann, stable . , 1980 Kobayashi, Lubke, Donaldson,
Hitchin, Uhlenbeck, Yau , Hermite-Einsteinstable . ,
. , Simpson Corlette Higgs. , D D
. Kobayashi-Hitchin.
, ., [53] . , [50] . .
16 . , Kobayashi-Hitchin, Kobayashi-Hitchin Simpson
, Kobayashi-Hitchin ., .
1 Narasimhan-Seshadri
Riemann , 1 ,1 . X Riemann .
1.1 .
• c1(L) = 0 X (L, ∂L).
• X π1(X) S1 := a ∈ C | |a| = 1 .
1.1 , ( ) ..
. X C∞- L , h , h ∇, (L, h,∇) . (L, h,∇) ,
X ∇ , π1(X) −→ S1 .ρ : π1(X) −→ S1 , ρ
. , π1(X) S1 . ( , X.) .
.
1.2 .
• X (L, ∂L) c1(L) = 0 .
• X (L, h,∇).
1
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X , T ∗X⊗C = Ω0,1X ⊕Ω1,0
X , X 1-form (1, 0)-form(0, 1)-form . , (L, h,∇)
∇ : C∞(X,L) −→ C∞(X,L⊗ T ∗X) = C∞(X,L⊗ Ω1,0)⊕ C∞(X,L⊗ Ω0,1)
∇0,1 +∇1,0 (0, 1)- (1, 0)- . , ∇0,1 L ., L Chern c1(L) 0 . , X
(L, h,∇) , c1(L) = 0 (L,∇0,1) . ,. h ∇ , ∇ ∇0,1 “
” ., , . ,
(E, ∂E) , E h , h , (0, 1)- ∂E
∇h = ∂E + ∂E,h . (E, ∂E , h) Chern . ChernR(h) . , R(h) ∈ C∞(X,End(E)⊗Ω1,1) , rankE = 1
R(h) ∈ C∞(X,Ω1,1) ., X (L, ∂L) c1(L) = 0 , R(h) = 0
h . L h0 . h0 eϕh0 ,R(eϕh0) = R(h0) + ∂∂ϕ . c1(L) = 0 ,
∫XR(h0) = 0 .
, ,∫X R(h0) = 0 , ∂∂ϕ = −R(h0) ϕ ,
. , X Kahler g , X f∫Xf = 0 , ∆Xϕ = f
ϕ . ,. , R(h) = 0 h
.
Narasimhan-Seshadri , 1.11.2 ? .
. , E, h, ∇ (E, h,∇) , (E,∇0,1). c1(E) = 0 .
, (E, ∂E) c1(E) = 0 ,. , Grothendieck , P1
⊕m∈ZOP1(m)rm
, (rm)m∈Z . P1 1 , rank 2O2
P1 , , Chern 0OP1(m) ⊕OP1(−m) (m ≥ 0) . , Chern 0
.P1 , π1(P
1) , (E, ∂E) O⊕ rankE
P1 . T π1(T ) Z2 , Atiyah,
. , 2 , ., stable, polystable . Narasimhan-Seshadri
.stability . X E , µ(E) := rank(E)−1
∫X c1(E) .
E 0 = E E µ(E) < µ(E) , Estable . “<” “≤” semistable .
E =⊕
Ei , i µ(E) = µ(Ei) , E polystable .Narasimhan-Seshadri 1965 [59] .
1.3 (Narasimhan-Seshadri) c1(E) = 0 E, E polystable . , stable
.
, ,, . 1960
. stability , Mumford [57]. , .
2
―68―
,,
. , Hausdorff , ., ,
. , Riemannstability . Mumford , stable
, semistable . (.) ,
,.
Narasimhan-Seshadri Narasimhan Seshadri ,Donaldson [12] . , Donaldson
.(E, ∂E) , (E, h,∇) polystable ,
. E ⊂ E E . h E
h , Chern R(h) . , π E E
. , TrR(h) = Tr(R(h)π
)−∣∣∂Eπ
∣∣2 (Chern-Weil ). ,
∫
X
c1(E) =
∫
X
Tr(R(h)) =
∫
X
Tr(πR(h))−∫ ∣∣∂Eπ
∣∣2 = −∫
X
∣∣∂Eπ∣∣2
. , Chern , 3R(h) = 0 . , µ(E) ≤ 0 . µ(E) = 0 , ∂Eπ = 0
. , E E⊥ , E = E ⊕ E⊥
. E ∇ E E⊥ , E, E⊥ ., E polystable .
. , polystable (E, ∂E) , R(h) = 0 h, . , ∆Xϕ = g
, R(h0) + ∂E(b−1∂E,h0b) = 0
, .X Kahler gX . , (1, 1)-form (0, 0)-form
Λ . (E, ∂E) h , End(E)⊗Ω1,1 R(h), End(E) ΛR(h) . h .E h0 . , det(E) det(h0)R(det(h0)) = 0 . Donaldson
h−1t
∂ht
∂t= −
√−1ΛR(ht) (1)
, C∞ 0 ≤ t < ∞ . ,. ,
(∂t +∆X)|ΛR(ht)|2 = −
∣∣∂ΛR(ht)∣∣2 ≤ 0
. , t sup |ΛR(ht)|2 ,. , ht ,
hti (ti → ∞) , h∞ R(h∞) = 0 . ht
., .
, Donaldson “Donaldson functional” . E HE
, h tangent space ThHE . b ∈ ThHE ,√−1
∫X Tr(bR(h))
, HE 1-form Φ . Donaldson Φ exact 1-form ., h(1) h(2) path Φ , M(h(1), h(2))
3
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. ddtM(h0, ht) = −
∫|F (ht)|2 ≤ 0 ,
M(h0, ht) ≤ 0 ., (E, ∂E) stable , Donaldson functional
. Simpson [68] .
1.4 (Donaldson, Simpson) (E, ∂E) stable . E h(1) ,C1, C2 > 0 .
dHE (h(1), h(2)) ≤ C1 + C2M(h(1), h(2)) (∀h(2) ∈ HE , det(h(2)) = det(h(1))).
, dHE HE .
, M(h0, ht) ≤ 0 , dHE (h0, ht) ,( ) hti (ti → ∞) .
( 1.4) . , C1, C2
, stability ,, C1, C2 , effective .
2 Kobayashi-Hitchin
Narasimhan Seshadri ,. , . Donaldson [15], Uhlenbeck-Yau [76]
.
Stability X . L ample . E X. rank(F ) > 0 OX - F , µL(F ) = rank(F )−1
∫c1(F )c1(L)
dimX−1
. , F ⊂ E 0 < rankF < rankE , µL(F ) < µL(E), E stable . , L . stable bundle
E =⊕
Ei , µL(E) = µL(Ei) polystable . stability Takemoto[73, 74] , slope stability Mumford-Takemoto stability
. stability , HilbertGieseker-Maruyama stability , 21
Bridgeland stability . , stability.
Hermitian-Einstein Kobayashi-Hitchin stability, Kobayashi [34] Hermitian-Einstein . X Kahler
ω , c1(L) . ω Λ . ,(E, ∂E) h Hermitian-Einstein .
ΛR(h) =TrΛR(h)
rankEidE . (2)
, ΛR(h) . TrR(h) det(E) ,. , r A r−1 Tr(A)Ir
0 A⊥ = A− r−1 Tr(A)Ir . ,A⊥ . Hermitian-Einstein ΛR(h)
ΛR(h)⊥ 0 , .. , Hermitian-Einstein
. , Hermitian-Einstein ,Kobayashi-Lubke , .
2.1 (Kobayashi-Lubke) Kahler (E, ∂E) Hermitian-Einstein, (E, ∂E) polystable.
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1 Chern-Weil . , E ⊂ E, E π , C > 0 ,
C rank(E)µL(E) =
√−1
∫Tr(πΛR(h))−
∫|∂π|2 ≤ rankE
rankE
√−1
∫TrΛR(h) = C rank(E)µL(E)
. , ΛR(h)⊥ = 0 , Tr((R(h)⊥)2
)ωn−2 ≥ 0 ,∫
X c1(E)2c1(L)dimX−2
∫X ch2(E)c1(L)
dimX−2 , Kobayashi-Lubke. (stable Bogomolov .)
, . , R(h) ∂∂.√−1Λ∂∂ ( )Laplacian , (2) h
. , 80 ,(2) .
, , , Kobayashi ,, stability Hermitian-Einstein ..
2.2 (Donaldson, Uhlenbeck-Yau) X . E stable, Hermitian-Einstein .
Donaldson [14], Donaldson [15], Uhlenbeck-Yau [76]. Uhlenbeck-Yau Kahler .
Kobayashi-Lubke , stability , Hermitian-Einstein. Kobayashi-Hitchin .
Kobayashi-Hitchin , , 4Donaldson [18] , . Donaldson
2 Donaldson .4 Riemann (E, h) ∇ , ∗F (∇) + F (∇) = 0
. , ,4
. , KahlerHermitian-Einstein . stable
, Gieseker Maruyama ,, .
Hitchin , stability Hermitian-Einstein ,Kobayashi-Hitchin ( Hitchin-Kobayashi ) .
, [14] , Narasimhan-Seshadri Hitchin Kobayashi, . ( ,
[41] .) Hermitian-Einstein Kobayashi , stability, “Kobayashi”
. , Kobayashi [25] , Hitchin ., 1979 9 “Non-linear problems in geometry”
, Kobayashi-Hitchin . , “Kahler (X,ω) E c1(E) = 0 , ω stable ,
ωdimX−1R(h) = 0 E h ” . ΛR(h) = 0. , “ , ”
. ( [25], [38] .) c1(E) = 0 ,. , Donaldson
.
, Narasimhan-Seshadri. , .
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2.3 (Donaldson, Uhlenbeck-Yau, Mehta-Ramanathan) X ..
• X ( X ).
• X polystable E , chi(E) = 0 (i > 0) .
, (E, h,∇) (E,∇0,1) .Chern . polystable , Chern-Weil .
, stable chi(E) = 0 (i > 0) (E, ∂E) , R(h) = 0 h ., . , rankE = 1 ,
E h0 R(h0) ∈ C∞(X,Ω1,1) (1, 1)- . Chern 0R(h0) 0 . , R(h0e
ϕ) = R(h0) + ∂∂ϕ ,, Kahler ∂∂- , R(h0) = ∂∂ϕ ϕ
, h = h0eϕ , R(h) = 0 .
, . ,. R(h) = 0
, . , R(h) = 0. , Hermitian-Einstein ΛR(h) = 0 .
, Kobayashi-Hitchin . , c1(E), ch2(E)∫Tr
((R(h)⊥)2
)ωdimX−2
(Kobayashi-Lubke ) , c1(E) = ch2(E) = 0 R(h) = 0. , Chern ,
stable . , “ ” ,Hermitian-Einstein . (§3–4) .
, Mehta-Ramanathan [43, 44] , 2.3 2, 3 Kobayashi-Hitchin
. , X stable ample generic , stable. Mehta-Ramanathan Donaldson 2
Kobayashi-Hitchin , .
3 Higgs Kobayashi-Hitchin
Riemann Higgs Donaldson, Uhlenbeck-Yau , Kobayashi-Hitchin. ,
. . ,. , , stability
Hermitian-Einstein , .Higgs Kobayashi-Hitchin .
Higgs X , (E, ∂E) X . θ (E, ∂E) Higgs ,, End(E)⊗Ω1,0
X , θ ∧ θ = 0 . E h , Chern
∇h = ∂E + ∂E,h , θ θ†h ∈ C∞(X,End(E)⊗ Ω0,1X ) .
3.1 D1 := ∇h+θ+θ†h , h (E, ∂E , θ) , (E, ∂E , θ, h) .
, ∂Eθ†h = R(h)+ [θ, θ†h] = ∂E,hθ = 0 . dimX = 1 , R(h)+ [θ, θ†h] = 0 Hitchin.
Hitchin Riemann , , Hitchin [23]
. R4 E, h, ∇ (E, h, ∇), . .
, t(x1, x2, x3, x4) = (x1 + t, x2, x3, x4) (E, h, ∇) .
, Rx2,x3,x4 E, h, ∇ . Lx1 ∇
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φ = ∇x1 − Lx1 , E φ
. , F (∇) + ∗F (∇) = 0 , (E, h,∇, φ) BogomolnyF (∇) − ∗∇φ = 0 . , 3 (E, h,∇, φ). . , 3 Nahm . Nahm
, “Nahm ” , Nahm.
Hitchin 2 . R4 = Cz × Cw , Cz- (E, h, ∇) . Cw
(E, h,∇) . , ∇z = Lz + f , ∇z = Lz − f † . (E,∇, f, f †), R(∇) +
[f dw, f †dw
]= 0 . , Hitchin .
,, Hitchin . (E,∇0,1, f dw, h) , w
. , Riemann Higgs (E, ∂E , θ) h. , 2 Riemann ,
Hitchin .
( Riemann , 2 ) Hitchin 2 ., Higgs .
3.2 (Hitchin, Donaldson, Diederich-Ohsawa) Riemann .
• 2 .
• 2 Higgs (E, ∂E , θ) , c1(E) = 0 .
• 2 (E,∇).
Higgs (E, ∂E , θ) stable , 0 = F E θ(F ) ⊂ F ⊗ Ω1
, µ(F ) < µ(E) . Riemann , µ(E) = 0polystable Higgs , 2 Hitchin [23] .
Higgs (E, ∂E , θ) h Hitchin , ∇ = ∇h + θ + θ† ., ( ) , (Higgs , ) ( , )
, . Riemann ,(V,∇) . h , ∇ = ∇h +Φh . , ∇h
, Φh End(V )⊗ (Ω1,0 ⊕Ω0,1) . , ∇h = ∂V,h + ∂V,h, Φh = θh + θ†h. ∂V,hθh = 0 , h . ( Kahler .)
, “ ” . .Riemann C∞- f : M1 −→ M2 df ∈ Hom(TM1, f
∗TM2) , TM1, f∗TM2
. f E(f) :=∫M1
|df |2 ( ) .
Riemann , . M1, M2
M2 , Eelles-Sampson.
(V,∇) h , X X H Fh .H , . X Kahler , “Fh ”
. , X , dimX = 1Kahler . , h Fh
.Donaldson[16] Eelles-Sampson , 2
. Diederich-Ohsawa[11] , PSL(2,R).
( ) 3.2 , , , Corlette [9]Simpson [68] . , ,
.
3.3 , .
7
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• .
• polystable Higgs (E, ∂E , θ) , chi(E) = 0 (i > 0) .
• (E,∇).
(Corlette) Corlette Riemann, . Donaldson Hitchin ,
. , Eelles-Sampson Riemann.
Corlette , Siu Bochner ., , Siu Bochner, .
, Corlette ,, Kahler .
Kahler , (V,∇) h , ∇ = ∂V,h+∂V,h+θh+θ†h, h (V,∇) , Λ(∂V,hθh) = 0 . ∂
2
V,h = ∂V,hθh =θ2h = 0 , h . Corlette
∂∂〈θ, θ〉ωn−2 =(C1|θ2h|2 + C2|∂V,hθh|2
)ωn (3)
( , C1, C2 > 0) . ,
, θ2h = ∂V,hθ = 0 . , ∂2
V,h = 0 ., , Corlette . ,
Kahler .
, ∂2
V,h = ∂V,hθh = θ2h = 0 , ∂V,hθh = 0 ,
∂2
V,h = 0, θ2h = 0 . , (V,∇, h) ∂V,h, ∂V,h, θh,
θ†h , ∂V,hθh = 0 ([47] [48] ), ∂V,hθh = 0 (3)
θ2h = 0 , ∂2
V,h = 0 .
Higgs Hodge (Simpson) , Simpson Higgs Kobayashi-Hitchin . Hitchin . , Simpson Hodge
, Kobayashi-Hitchin Hitchin.
f : X −→ Y , Hk(f−1(y))(y ∈ Y ) , Y .Griffiths Hodge . Hodge Griffiths Deligne
. Deligne Hodge ., Hodge ,
. , Hodge ,.
( ) Hodge LC, LC⊗OX Hodge F , LC⊗OX† HodgeG, 〈·, ·〉 . , LC ⊗OX , LC ⊗OX†
Griffiths , ., E = GrF (LC⊗OX) , θ : GriF −→ Gri−1
F ⊗Ω1X Higgs θ ∈ End(E)⊗Ω1
. (E =⊕
Ei, θ) , , “ ” .∇ = ∂E + ∂E,h + θ+ θ† Simpson , (E, ∂E , θ)
Higgs , , h F (h) = (∇h + θ + θ†)2 = 0. TrF (h) = 0 1 ,
ΛF (h)⊥ = 0 , . ,, ΛF (h)⊥ = 0 Hermitian-Einstein .
Simpson , Donaldson, Uhlenbeck-Yau , Higgs Kobayashi-Hitchin. , Higgs Hermitian-Einstein , Kobayashi-Lubke ,
8
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Bogomolov-Gieseker . .
∫
X
(2c2(E)− r − 1
rc1(E)2
)ωn−2 = C
∫
X
tr(F (h)⊥F (h)⊥)ωn−2 ≥ 0.
, F (h)⊥ := F (h) − TrF (h)rankE idE . , ,
Hermitian-Einstein . , (E, θ, h)E =
⊕Ei , θEi ⊂ Ei−1 , ( ) Hodge
. , Hodge Kobayashi-Hitchin .Simpson , . , Higgs
Hermitian-Einstein . ([70] .), Λ(∂θ) = 0 Hermitian-Einstein . ,
Higgs . ,Kobayashi-Hitchin , .
3.2 3.3 . Hitchin Simpson ,hyperkahler Hodge .
, .
4 ( )
4.1 1
, . 1 , Riemann, , Higgs .
Mehta Seshadri [45] , Simpson [69].
(E, h,∇) C \D . (E,∇0,1) C \D. D = ∅ , C \D , (E,∇0,1)
. C , D.Delinge . . Y = z ∈ C | |z| < 1
, Y \ 0 (E, h,∇) . , E v A, ∇v = vAdz/z . , A α 0 ≤ Re(α) < 1 .
, v E Y E . E Deligne .. , E|0 Res(∇) ,
E|0 =⊕
EαE|0 . , −1 < a ≤ 0 , Fa(E|0) =⊕
−Re(α)≤a EαE|0
, . ,.
, , U ⊂ Y , 0 ∈ U ,PaE(U) U \ 0 E , |s|h = O(|z|−a−ε) (ε > 0) . ,0 ∈ U , PaE(U) U E . , OY - PaE . ,
OY - . P0(E) = E , Pa(E)|0 −→ P0(E)|0 Fa
. .C \D , P ∈ D , C
E , E|P (P ∈ D) . E∗
. , deg(E∗) µ(E∗) .
deg(E∗) :=
∫
X
c1(E)−∑
P∈D
∑
−1<a≤0
a rankGrFa (E|P ), µ(E∗) :=deg(E∗)
rank E.
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E1 ⊂ E , Fa(E1|P ) := Fa(E|P ) ∩ E1|P , E1 ., stability . , Mehta-Seshadri
.
4.1 (Mehta-Seshadri) , .
• C \D .
• (C,D) polystable E∗ µ(E∗) = 0 .
Maruyama Yokogawa [42, 77] , stablestable Higgs .
,. , ,
, , ,.
, Simpson [69]. , Y \0 . (E, ∂E , θ, h)
Y \ 0 . θ = f dz/z . det(t idE −f) =∑
aj(z)tj
. aj(z) z = 0 , .P∗E , (E, ∂E , θ) OY - P∗E := (PaE | a ∈ R)
. , E1 := (E, ∂E + θ†) h , OY - P∗E1
. Simpson [69] [10] .
4.2 (Simpson)
• (P∗E, θ) Higgs . , PaE OY - , θPaE ⊂PaE ⊗ Ω1(logO).
• (P∗E1,D1) . , PaE1 OY - , D1PaE1 ⊂PaE1 ⊗ Ω1(logO).
, Simpson GrPa (E) := PaE/P<aE Res(θ) , GrPa (E1) := PaE1/P<aE1
Res(D1) .. Hodge Griffiths Schmid
([67] ), ,.
Simpson Riemann C D, (C,D) , C \D (E, ∂E , θ, h) , D
. P ∈ D , C \D , (C,D)Higgs (P∗E, θ) (P∗E1,D1) .
, µ(P∗E) = µ(P∗E1) = 0 . , polystable ., Chern-Weil µ(P∗E) µ(P∗E1) ,
, . , ., .
4.3 (Simpson) , .
• (C,D) .
• (C,D) Higgs (P∗E0, θ) , polystable µ(P∗E0) = 0.
• (C,D) (P∗E1,∇) , polystable µ(P∗E1) = 0.
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Kahler Kobayashi-Hitchin 4.3 , KahlerHermitian-Einstein . §3 , Simpson
, [68] Kahler .(X,ω) Kahler . . (i) , (ii) C∞- ϕ : X −→ R≥0 ,a ∈ R ϕ−1(0 ≤ t ≤ a) , ∆Xϕ , (iii) a : R≥0 −→ R≥0 ,
a(0) = 0, a(t) = t (t ≥ 1) , .
• f : X −→ R≥0 , ∆Xf ≤ B B . , BC(B) , sup |f | ≤ C(B)a(
∫f).
, ∆Xf ≤ 0 , ∆Xf = 0., Simpson analytic stability . (E, ∂E , θ) X Higgs , h0
E . |ΛF (h0)|h0 . ,
deg(E, h0) :=√−1
∫
X
Tr(ΛF (h0)
)dvolX
. , F ⊂ E . 2 Z(F ) ⊂ X , Z(F )F E . , F|X\Z(F ) h0,F . ,
deg(F, h0) :=√−1
∫
X\Z(F )
Tr(ΛF (h0,F )
)dvolX ∈ R ∪ −∞
. F ⊂ E θ(F ) ⊂ F ⊗ Ω1 0 < rankF < rankE ,deg(F, h0)/ rank(F ) < rank(E, h0)/ rankE , (E, ∂E , θ, h0) analytically stable .Simpson [68] .
4.4 (Simpson) (E, ∂E , θ, h0) analytically stable . , (E, ∂E , θ) Hermitian-Einstein(i) det(h) = det(h0), (ii) h h0 , (iii) ∂(h0h
−1) ∈ L2 .
. Simpson , G X , (E, ∂E , h0), G . (
, 6.1 .) Higgs , .
4.2
[53] .
X , D . (E, ∂E , θ, h) X \D. Higgs (E, ∂E , θ) T ∗(X \D) . Higgs
, Σθ . T ∗X(logD) Σθ X , (E, ∂E , θ, h)(X,D) .
. (U, z1, . . . , zn) U ∩D =⋃
i=1zi = 0, θ =
∑i=1 fi dzi/zi +
∑nj=+1 gjdzj . det(t id−fi) =
∑ai,kt
k,
det(t id−gj) =∑
bj,ktk . (E, ∂E , θ, h)|U\D (U,U ∩D) ,
ai,k, bj,k U .
λ , (E, ∂E + λθ†h) Eλ . OX\D-
. , λ- Dλ := ∂E + λθ† + λ∂E + θ . 1 ,, X \D (Eλ,Dλ) (X,D) .
X = ∆n, D =⋃
i=1zi = 0 . a ∈ R , U ⊂ X , PaEλ(U)
U \D Eλ section s , P ∈ U ∩D , |s|h = O(∏
i=1 |zi|−ai−ε)(∀ε > 0)
. , OX -module PaEλ . ([47] ).
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4.5 PaEλ OX- . , v , vi bi ∈ R , c ∈ R
, .
PcEλ =
n⊕
i=1
O([c− bi])vi.
( , e = (e1, . . . , e) ∈ R , [ei] := maxn ∈ Z |n ≤ ei, [e] = ([e1], . . . , [e]) .), c ∈ R DλPcEλ ⊂ PcEλ ⊗ Ω1(logD) .
Kobayashi-Hitchin X , D . D =⋃
i∈ΛDi
. (X,D) (E, ∂E , θ, h) , a ∈ RΛ , D4.5 , PaEλ OX - . (PaEλ
∣∣a ∈ RΛ)P∗Eλ . , a DλPaEλ ⊂ PaEλ ⊗ Ω1
X(logD) . (P∗Eλ,Dλ)λ- . ( , λ = 1 , λ = 0
Higgs .)X , L ample . 4.5 OX -
P∗F , chi(P∗F) . (i = 1, 2 [46] . [26, 27] .), P∗Eλ , ch1(P∗Eλ) = 0,
∫X ch2(P∗E)c1(L)dimX−2 = 0
([46, 48] ). Chern vanishing , D, [47] , [47]
. , µL(P∗Eλ) = (rankE)−2∫X c1(P∗Eλ)c1(L)
dimX−1 , polystable([46, 48] ). ,
λ (P∗Eλ,Dλ) . .
4.6 X , D , .
• (X,D) .
• (X,D) λ- (P∗V ,Dλ) , polystable ,
ch1(P∗V) = 0,
∫
X
ch2(P∗V)c1(L)dimX−2 = 0
.
, Simpson Kobayashi-Hitchin ( 4.4)Hermitian-Einstein , . ,
Gr ResDi(Dλ) GrF ResDi(D
λ), 4.4 h0 , 4.4
. 2 , 4.4 , Di Dj , GrF ResDi(Dλ)
GrF ResDj (Dλ) .
. , ,.
, F(ε) , GrF
(ε)
ResDi(Dλ)
. , 4.4 h0 , F (ε)
Hermitian-Einstein h(ε) . , limε→0 h(ε) ,
., ε → 0 ,
. , ., F
(ε) → F(0) , Hermitian-Einstein
h(ε) → h(0) . , h(ε)
. 4.4 h(ε) , F (ε) h0,ε , (i) h0,ε → h0,0,(ii) dH(h0,ε, h) ≤ C1 + C2M(h0,ε, h), .
, M(h0,ε, h(ε)) ≤ 0 , dH(h0,ε, h
(ε)) ≤ C1 .
12
―78―
, “θ residue ”. , . Corlette
. ([48] . , Corlette , [29], [47]. [29] , , [47]
.) θ residue ,, Deligne residue ,
Chern 0 , polystable .
, . , N > 0, ΣE T ∗X(logD)⊗OX(ND) X . , ΣE D
. ., Kobayashi-Hitchin . λ = 1 [51] . λ = 0
, [46, 48, 51] . dimX = 1 Biquard-Boalch [3], Sabbah [63].
, “Higgs residue” ,
√−1R- . ,√
−1R- .(V,∇) (X,D) , P ∈ D , , D
, ., ,
([32, 33], [49, 51]). , ., [49, 51] . 2 ,
“ ” ([49] )., p > 0 , p- Higgs ,
, . , 2 , Kobayashi-Hitchin. 2 , ,
. , ,Higgs ,
, .
D D , D-, . ([64, 65], [47, 51] .) Simpson “ ”
, Hodge [66] ., D , D D
.
5
λ λ- , λ-Kobayashi-Hitchin .
, Kobayashi-Hitchin ([54] ). .
5.1
GCK S1 ×C dt dt+ dw dw . U ⊂ S1 ×C
, E, h, ∇, φ , Bogomolny ,
F (∇)− ∗∇φ = 0
, (E, h,∇, φ) U . , F (∇) ∇ , ∗ Hodge.
13
―79―
Z ⊂ S1 × C . (S1 × C) \ Z monopole (E, h,∇, φ) , GCK(generalized Cherkis-Kapustin) .
• P ∈ Z Dirac ([31] ). , Dirac 1. , P Q |φ|Q|h = O
(d(P,Q)−1
)([56]
).
• |w| → ∞ , F (∇) → 0 |φ| = O(log |w|).S1 × C , Cherkis Kapustin [7, 8] .
Nahm , C∗ genericity .
C[β1] Φ∗ Φ∗(β1) = β1 + 2√−1λ . C[β1]- V C-
Φ∗ : V −→ V , f ∈ C[β1] s ∈ V , Φ∗(fs) = Φ∗(f) ·Φ∗(s)(V ,Φ∗) (2
√−1λ-) . A :=
⊕n∈ZC[β1](Φ
∗)n , Φ∗ · f = Φ∗(f) ·Φ∗
A , A- . V (i) A-, (ii) dimC(β1) V ⊗C[β1] C(β1) < ∞ , .λ- , .
, “ ” . (.) .
V . V(V,m, (t1,x,Lx |x ∈ C)) .
• C[β1]- V ⊂ V , A · V = V V ⊗ C(β1) = V ⊗ C(β1) .
• m : C −→ Z≥0 ,∑
x∈Cm(x) < ∞ , D := x ∈ C |m(x) > 0 , V [∗D] =(Φ∗)−1(V )[∗D] . ( , [∗D] β1 − x (x ∈ D) .)
• x ∈ C , t1,x := (0 ≤ t(1)1,x < · · · < t
(m(x))1,x < T ). ( , m(x) = 0 .)
• x ∈ C , V [∗D] ⊗ C[[β1 − x]] Lx :=(Lx,i
∣∣ i = 1, . . . ,m(x) − 1). Lx,0 :=
V ⊗ C[[β1 − x]], Lx,m(x) := (Φ∗)−1(V )⊗ C[[β1 − x]] .
∞ β−11 C[[β−1
1 ]] , β−11
Laurent C((β−11 )) . V V ∞
. , V := V ⊗C[β−1
1 ] C[[β−11 ]]. C((β−1
1 ))- .
V , V P∗V =(PaV
∣∣ a ∈ R),
, V C[[β−11 ]]- .
• PaV ⊂ PbV (a ≤ b).
• a ∈ R, n ∈ Z , Pa+nV = βn1PaV .
• a ∈ R , ε > 0 , PaV = Pa+εV .
P∗V ..
V C- Φ∗ , V C- . , V ⊗ C((β−1/p1 )) C-
. Φ∗ . , f ∈ C((β−1/p1 )) s ∈ V ⊗ C((β
−1/p1 ))
, Φ∗(fs) = Φ∗(f) · Φ∗(s) . , (V ,Φ∗) C((β−11 )) .
, Turrittin [75] . ([6], [19], [62] .)
. p ∈ Z>0 , S(p) :=∑p−1
j=1 bjβ−j/p1
∣∣ bj ∈ C
.
, p ∈ Z>0 ,
V ⊗ C((β−1/p1 )) =
⊕
ω∈Q
⊕
α∈C∗
⊕
b∈S(p)
V p,ω,α,b (4)
14
―80―
, V p,ω,α,b Φ∗ , Lp,ω,α,b ⊂ V p,ω,α,b ,
(α−1β−ω1 Φ∗ − (1 + b) id)Lp,ω,α,b ⊂ β−1
1 Lp,ω,α,b
.V P∗V , V ⊗ C((β
−1/p1 ))
P∗(V ⊗ C((β−1/p1 ))) , .
5.1 V P∗V ,.
• P∗(V ⊗ C((β−1/p1 ))) =
⊕ω,α,bP∗V p,ω,α,b .
• , a ∈ R ,(α−1β−ω
1 Φ∗ − (1 + b) id)PaV p,ω,α,b ⊂ β−1
1 PaV p,ω,α,b.
Stability (V , (V,m, t1,x,Lxx∈C),P∗V ) , . C[β1]- V
, P0V , OP1- P0FV . a ∈ R , C
GrPa (V ) := Pa(V )/P<a(V )
. (4) ω ∈ 1pZ ,
r(ω) :=∑
α,b
dimC((β
−1/p1 ))
V ω,α,b
. ω ∈ 1pZ , r(ω) = 0 . , m(x) > 0 , i = 1, . . . ,m(x) ,
deg(Lx,i, Lx,i−1) ,
deg(Lx,i, Lx,i−1) = length(Lx,i/Lx,i ∩ Lx,i−1
)− length
(Lx,i−1/Lx,i ∩ Lx,i−1
)
. , (V , (V,m, t1,x,Lx),P∗V )
deg(V , (V,m, t1,x,Lx),P∗V ) := deg(P0FV )−∑
−1<a≤0
a dimC GrPa (V )−∑
ω
ω
2r(ω)
+∑
x∈C
∑(1 − t
(i)1,x) deg(Lx,i, Lx,i−1) (5)
.
V := V ⊗ C(β1) V
Φ∗(V) ⊂ V
,
V := V ∩ V, V
:= A · V , Lx,i := Lx,i ∩ V , V
:= V
⊗ C((β−11 )), PaV
:= PaV ∩ V
, (V , (V ,m, t1,x,Lxx∈C),P∗V) .
5.2 C(β1)- V ⊂ V Φ∗(V
) = V
0 < dim
C((β−11 )) V
< dim
C((β−11 )) V
deg(V , (V ,m, t1,x,Lxx∈C),P∗V
)
dimC((β−1
1 )) V <
deg(V , (V,m, t1,x,Lxx∈C),P∗V )
dimC((β−1
1 )) V
, (V , (V,m, t1,x,Lxx∈C),P∗V ) stable . semistable polystable.
15
―81―
, ([54] ).
5.3 λ , .
• S1 × C GCK
• polystable 2√−1λ- , 0 .
Charbonneau Hurtubise [5] , S1 Riemann Σ Dirac, Σ “stability ” .
5.3 λ = 0 . “ ” λ = 0 5.3.
5.2
λ = 0 Z Rt × Cw , S1 × Cw .
κn(t, w) = (t+ n,w) (n ∈ Z).
(E, h,∇, φ) S1 × Cw . ..
• ∂E,w := ∇w , E|t×Cw.
• ∂E,t := ∇t−√−1φ (t, w) | 0 ≤ t ≤ 1 , F : E|0×Cw
E|1×Cw
.
• Bogomolny [∂E,t, ∂E,w] = 0 , F .
• E|0×Cw= E|1×Cw
.
, E|0×CwF , 0- H0(Cw, E|0×Cw
). , .
λ Rt × Cw (t0, β0), (t1, β1) .
(t0, β0) =1
1 + |λ|2((1− |λ|2)t+ 2 Im(λw), w + 2
√−1λt+ λ2w
),
(t1, β1) =(t0 + Im(λβ0), (1 + |λ|2)β0
)=
(t+ Im(λw), w + 2
√−1λt+ λ2w
).
Z- (t1, β1) .
κn(t1, β1) = (t1 + n, β1 + 2√−1λn) (n ∈ Z).
Rt1 × Cβ1 , S1t × Cw .
, dt0 dt0 + dβ0 dβ0 = dt dt + dw dw . , Bogomolny , E∂E,t0 := ∇t0 −
√−1φ ∂E,β0
:= ∇β0. .
∂t1 = ∂t0 , ∂β1=
λ
1 + |λ|21
2√−1λ
∂t0 +1
1 + |λ|2 ∂β0.
, E ∂E,t1 ∂E,β1.
∂E,t1 := ∂E,t0 , ∂E,β1:=
λ
1 + |λ|21
2√−1λ
∂E,t0 +1
1 + |λ|2 ∂E,β0.
,[∂E,t1 , ∂E,β1
]= 0 . , λ = 0 .
16
―82―
• ∂E,β1E|t1×Cβ1
.
• ∂E,t1 , F : E|0×Cβ1 E|1×Cβ1
.
, E|0×Cβ1.
F : E|0×Cβ1 (Φ−1)∗(E|0×Cβ1
), (Φ(β1) = β1 + 2√−1λ).
, 2√−1λ- H0(Cβ1 , E|0×Cβ1
) . , .
t1 × Cβ1 ⊂ P1β1
. U ⊂ P1β1
(∞ ∈ U) ,
P(E|t1×Cβ1)(U) :=
s ∈ Γ(U \ ∞, E)
∣∣ |s|h = O(|β1|N ) ∃N ∈ R,
Pa(E|t1×Cβ1)(U) :=
s ∈ Γ(U \ ∞, E)
∣∣ |s|h = O(|β1|a+ε) ∀ε > 0,
, OP1(∗∞)- P(E|t1×Cβ1) OP1- Pa(E|t1×Cβ1
) (a ∈ R) .
5.4 (E, h,∇, φ) GCK ,
• P(E|t1×Cβ1) OP1(∗∞)- .
• Pa(E|t1×Cβ1) (a ∈ R) OP1- .
• F P(E|0×Cβ1) P(E|1×Cβ1
) .
( , F Pa(E|0×Cβ1) Pa(E|1×Cβ1
) .)
S1t × Cw GCK (E, h,∇, φ) ,
C[β1]- V = V := H0(P1,P(E|0×Cβ1
))
. , V 2√−1λ-
.
H0(P1,P(E|0×Cβ1
)) F H0
(P1,P(E|1×Cβ1
))= H0
(P1, (Φ−1)∗P(E|0×Cβ1
))= H0
(P1,P(E|0×Cβ1
)).
, (Pa(E|0×Cβ1| a ∈ R) V P∗V . ,
., (E, h,∇, φ) Z Dirac , F . V := V ⊗ C(β1)
, V = A · V . V ∞, .
5.3 .
Hodge C2 = (z, w) dz dz + dw dw , hyperkahler. U ⊂ C2 . λ , Uλ .
U (E, h,∇) , λ , Uλ
Eλ . Γ ⊂ C2 , U Γ- , (E, h,∇) Γ- , Eλ Γ- .Γ = Cz × 0, U = Cz × Uw , Γ- Uw . §3
Hitchin . , λ , Uλ Γ- Uw λ-. , Γ- Γ- , P1
λ- .Γ = (R×
√−1Z)× 0, U = Cz ×Uw , Γ- S1 ×Uw .
, Γ- (E, ∂E,t0 , ∂E,β0) . ,
, Hodge .
17
―83―
5.3
S ⊂ C : S −→ Z>0 , P (y) :=∏
a∈S(y − a)(a) ∈ C(y) . V P :=C(y)e1 ⊕ C(y)e2 , Φ∗ .
Φ∗(e1, e2) = (e1, e2)
(0 P (y)1 0
).
VP := C[y]e1 ⊕C[y]e2 , 0- V P := A · VP ⊂ V P . 5.3 ,V P .
5.5 (ta)a∈S ∈ 0 ≤ x < 1S , :=∑
a∈S (a) = degy P .
( ) 0- V P (S1 × C) \ (ta, a) | a ∈ S .
( ) 0- V P (S1 × C) \ (ta, a) | a ∈ S ,
(d1, d2) ∈ R2
∣∣∣ d1 + d2 +∑
a∈S
(1− ta)(a) = 0
.
5.3 , V P stable 0 0 .Φ∗ C , .
5.6 V
V P = V P ⊗ C(y) C(y)- , Φ∗(V) = V
, V
V P 0.
, V P 0- 0 , stability.
, .
• m : C −→ Z≥0 , m(a) = 1 (a ∈ S), m(a) = 0 (a ∈ S) .
• a ∈ S , 0 ≤ ta < 1 .
, ∞ , 0 .
. τ ∈ C((y−1/2)) τ2 = P (y) y−1/2τ ∈ C[[y−1/2]]. v1 = τ e1 + e2, v2 = τ e1 − e2 , Φ∗(v1) = τ v1, Φ
∗(v2) = −τ v2.
5.7 P∗V , degP(v1) = degP(v2) =: d . ( , degP(u) :=mina ∈ R |u ∈ Pa.)
e1 = (2τ )−1(v1 + v2), e2 = 2−1(v1 − v2) , degP(e1) =d−2 , degP(e2) =
d2 . ,
0- ,
0− d−
2− d
2+
∑
a∈S
(1 − ta)(−(a))− 1
2(/2)2 = −d−
∑
a∈S
(1 − ta)(a)
. 0 d . , 5.5 . 2k , τ ∈ C((y−1)) τ2k = P (y) y−1τ ∈ C[[y−1]] . v1 = τke1 + e2,
v2 = τke1 − e2 Φ∗(v1) = τkv1, Φ∗(v2) = −τkv2 Φ∗ . , ∞
P∗V , degP(vi) = di (i = 1, 2) .
k − d1 − d2 −∑
a∈S
(1− ta)(a)−1
2(k) · 2 = −d1 − d2 −
∑
a∈S
(1− ta)(a)
, 5.5 .
18
―84―
6 -Hitchin
Simpson 5.3 , 4.4 .(X, gX) n , G .
ϕ : X −→ R>0 G . , (i) ϕ(P ) → 0 (P → ∞), (ii)∫Xϕ < ∞ .
C1, C2 > 0 .
- f : X −→ R≥0 bounded , ∆Xf ≤ Bϕ B > 0 . ,supP∈X f(P ) ≤ C1B + C2B
∫fϕ .
, ∆Xf ≤ 0 , ∆Xf = 0 .
(E, ∂E) X G , h0 E , |ΛF (h0)|h0 < Bϕ . (E, ∂E , h0)G- analytically stable , G- E ⊂ E 0 < rankE < rankE
,deg(E, h0)
rankE<
deg(E, h0)
rankE
. ([55] ).
6.1 (E, ∂E , h0) G- analytically stable , E G- Hermitian-Einsteinh . (i) det(h) = det(h0), (ii) h h0 , (iii) ∂E(h · h−1
0 ) L2.
C2z,w Z2- κ , R- ρ .
κ(n1,n2)(z, w) = (z + n1 +√−1n2, w), ρs(z, w) = (z + s, w).
X := C2/Z2 = S1 × (S1 × Cw) . X S1- ρ . q0 : X −→ S1 × C
(z, w) −→ (Im(z), w) = (t, w) . §5.2 ,.
6.2 Z ⊂ S1 × C .
• X \ q−10 (Z) S1- (E, ∂E) , (S1 × C) \ Z E
(∂E,t, ∂E,w) .
• X \ q−10 (Z) S1- , (S1 × C) \ Z .
X Xλ .
α0 :=1
1 + |λ|2 (z + λw − λw + |λ|2), β0 :=1
1 + |λ|2 (w + λ(z − z) + λ2w).
dα0 dα0 + dβ0 dβ0 = dz dz + dw dw . , (t0, β0) = (Im(α0), β0) . .
6.3 Z ⊂ S1 × C .
• Xλ\q−10 (Z) S1- , (S1×C)\Z (∂E,t0 , ∂E,β0
)
.
• Xλ \ q−10 (Z) S1- , (S1 × C) \ Z .
, analytically stable Kobayashi-Hitchin ( 6.1) , (S1 ×C) \Z.
19
―85―
Hermitian-Einstein Hermitian-Einstein . , Donaldson [13]
, C2 L2- , (P2, H∞) .P2 , . ,
Bando ALE- [1], Jarvis R3 [28] .Ni [60], Ni-Ren [61] Kahler ,
[58] ., (E, ∂E) , analytically stable
Hermitian-Einstein , . ( , Donaldson.) , Laplacian , ,
., analytically stable Hermitian-Einstein ,
, . , (R2 ×C
) Kobayashi-Hitchin ([4], [52] ) 6.1 .
Hermitian-Einstein 6.1 , Donaldson Hermitian-EinsteinDirichlet [17] . , stability .
6.4 (Donaldson) X Kahler , (E, ∂E) X .E|∂X C∞ h∂X , ΛR(h) = 0, h|∂X = h∂X E h .
f|∂X = 0 , ∆ , 0, (∂t + ∆X)θ ≤ 0 θ : R ×X −→ R≥0 , supx∈X θ(t, x) ≤ Ce−µt
. h−1t ∂tht = −2
√−1ΛF (h) ht|∂X = h∂X , Et = |ΛF (ht)|2ht
(∂t + ∆X)Et ≤ 0 . , Et ≤ Ce−µt . ,∫ t
0
√Esds , ht h0 ,. , 0 , , stability
.
6.1⋃Zi = X Zi ⊂ X , (E, ∂E)|Zi
Hermitian-Einstein hi , hi|∂Zi
= h0|∂Zi. hi .
. Dirichlet Donaldson functional , M(h0|Zi, hi) ≤ 0
. , hi . , HermiteDonaldson functional , Donaldson Simpson
. , . ,Ni [60] .
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