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

―67―

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

―69―

. 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.

4

―70―

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. , .

5

―71―

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 ∇

6

―72―

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

―73―

• .

• 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

―74―

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.

9

―75―

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.

10

―76―

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] ).

11

―77―

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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