Phuong VTMinh_Kahler_manifolds

B GIO DC V O TO

TRNG I HC S PHM H NI

V Th Minh Phng

A TP KAHLER

KHA LUN TT NGHIP I HC

Chuyn ngnh: Hnh hc v Tp

Gio vin hng dn: PGS.TSKH. S c Quang

H NI, 2015

Kha lun tt nghip a tp Kahler

LI CM N

Kha lun c hon thnh di s hng dn, ch bo tn tnh ca PGS.TSKH.

S c Quang. Nhn dp ny ti xin by t lng bit n su sc n ngi Thy tn

tnh gip ti. Ti cng xin chn thnh cm n cc Thy, C phn bin thnh thi

gian c v gp nhng kin qu bu cho kha lun ny.

Cui cng ti xin chn thnh cm n cc Thy, C trong Khoa Ton - Tin Trng

i hc S Phm H Ni, c bit l cc Thy, C trong B mn Hnh hc, cc bn

sinh vin v ngi thn gip ti sut nhng nm thng hc tp v nghin cu ti

trng.

H Ni, 01 thng 05 nm 2015

Tc gi kha lun

V Th Minh Phng

V Th Minh Phng 1

LI M U

a tp Kahler l mt a tp m trn c trang b ba cu trc tng thch vi

nhau, l cu trc Riemannian, cu trc Hermit v cu trc symplectic. Hnh hc

nghin cu cc tnh cht xung quanh a tp Kahler c gi l hnh hc Kahler,

c pht trin mnh m t nhng nm 70 ca th k trc bi nhiu nh ton hc nh

Alan Weinstein, Mikhail. L Gromov, Clifford Taube.... Hin nay a tp Kaler ang l

i c nghin cu trong nhiu ngnh ton hc.

Mc ch ca chng ti trong kha lun ny l i xy dng khi nim, a ra mt

s v d in hnh v a tp Kahler v cui cng l chng minh mt s tnh cht c

trng ca a tp Kahler.

V thi gian v kin thc c hn, nn kha lun khng th trnh khi nhng thiu

st. Chng ti rt mong nhn c nhng ng gp qu bu t pha bn c.

2

Mc lc

LI CM N 1

LI M U 2

MC LC 3

1 Cu trc phc v cu trc Hermit 4

1.1 Cu trc hu phc trn khng gian vc t thc . . . . . . . . . . . . . . 4

1.2 i s ngoi trn khng gian vc t thc . . . . . . . . . . . . . . . . . . 6

1.3 Dng vi phn trn Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Php tnh vi phn trn a tp phc 15

2.1 Cu trc hu phc trn a tp . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Dng vi phn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 a tp Kahler 21

3.1 Metric Kahler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 c trng ca a tp Kahler . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Phn th vc t Hermit v lin thng Chern . . . . . . . . . . . . 27

3.2.2 Cc tnh cht c trng ca a tp Kahler . . . . . . . . . . . . . 31

TI LIU THAM KHO 35

3

Chng 1

Cu trc phc v cu trc Hermit

Trong chng ny, chng ti trnh by s lc cc khi nim v tnh cht lin quan n

cu trc phc v cu trc Hermit. Cc chng minh chi tit c th xem trong [2].

1.1 Cu trc hu phc trn khng gian vc t thc

Cho V l mt khng gian vc t thc hu hn chiu v cho , l mt tch v hngtrn V .

nh ngha 1.1.1 (Cu trc hu phc trn khng gian vc t). Mt t ng cu

I : V V tha mn I2 = Id c gi l mt cu trc hu phc trn V .

Ta thy, nu V l C-khng gian vc t th trn V c cu trc hu phc t nhin xcnh bi:

I :V Vv 7 i.v.

B 1.1.2. Nu I l mt cu trc hu phc trn khng gian vc t thc V th V c

cu trc C-khng gian vc t cm sinh t I.

Tht vy, ta ch vic nh ngha php nhn vc t vi s phc bi: (a + bi).v =

a.v + b.Iv. Do , nu V chp nhn cu trc hu phc th V c s chiu thc chn.

nh ngha 1.1.3. Ta nh ngha VC := V R C.

4


Khi V c coi nh khng gian con ca VC bi ng nht v vi v 1. Php tonlin hp trn VC cho bi:

V R C V R C

v 7 v ,

Tc l v 7 v . D thy ngay V l khng gian con bt bin qua php lin hp.Khi nu I l cu trc hu phc trn V th I c m rng thnh t ng cu trn

VC (vn k hiu l I) bi I(v z) = I(v) z. Ta d dng nhn thy rng I c ng haigi tr ring l i.

Trong phn tip theo ca kha lun, nu khng c g ch c bit th thay v vit

I(v) ta s vit n gin l Iv, vi I l t ng cu v v l vc t.

nh ngha 1.1.4. Ta t:

V 1,0 = {v VC | Iv = i.v},

V 0,1 = {v VC | Iv = i.v},

tng ng l cc khng gian con ring ng vi tr ring i,i ca I.

Ta d thy rng VC = V1,0 V 0,1. Hn na c th chng minh c:

V 1,0 = {12

(v iIv) | v V },

V 0,1 = {12

(v + iIv) | v V },

v php lin hp phc trn VC cm sinh mt R-ng cu V 1,0 = V 0,1.

B 1.1.5. Cho V l khng gian vc t thc vi cu trc hu phc I. Khi khng gian

i ngu V = HomR(V,R) c cu trc hu phc cm sinh t I cho bi: I(f)(v) = f(Iv).Hn na ta c (V )C = HomR(V,C) = (VC) c cm sinh bi:

(V )1,0 = {f HomR(V,C) | f(Iv) = if(v)} = (V 1,0)

(V )0,1 = {f HomR(V,C) | f(Iv) = if(v)} = (V 0,1)

V Th Minh Phng 5


1.2 i s ngoi trn khng gian vc t thc

Gi s rng V l khng gian vc t thc chiu d. Ta t:

V =

dk=0

kV,

VC =

dk=0

kVC.

nh ngha 1.2.1. Cho I l cu trc hu phc trn V . Vi VC = V1,0 V 0,1 ta t:p,q

V =p

V 1,0

C

qV 0,1.

Khi ta c mnh sau.

Mnh 1.2.2. Vi cc k hiu nh trn, ta c

(i)p,qV l khng gian con ca p+q VC,

(ii)k VC =

p+q=k

p,q V,(iii) php lin hp phc trn

VC xc nh mt ng cu:p,qV =

q,pV,

(iv) ta c nh x: p,q:V

r,s p+r,q+sV(, ) 7 .

Gi s rng khng gian vc t V c chiu thc d. Khi ta c cc php chiu chnh

tc: k : VC k VC,p,q : VC p,q V.nh ngha 1.2.3. Cu trc hu phc I trn V c gi l tng thch vi tch v

hng , nu Iu, Iv = u, v. Ngha l I O(V, , ).

V Th Minh Phng 6


nh ngha 1.2.4. Cho (V, , ) l khng gian vc t Euclide hu hn chiu, cho I lcu trc hu phc tng thch vi tch v hng , . Khi dng c bn lin kt vi(V, , ) c nh ngha bi:

(u, v) = u, Iv = Iu, v. (V I2 = Id).

Ch rng ta c th m rng tch v hng , ti dng Hermit xc nh dng , C trn VC bi:

u , v = u, v.

B 1.2.5. Cho V l khng gian Euclide, I l cu trc hu phc tng thch th dng

c bn l dng thc c kiu (1, 1), ngha l 2 V 1,1 V .Chng minh. Trc tin ta c:

(u, v) = u, Iv = Iu, I2v = Iu, v = v, Iu = (v, u).

Vy nn 2 V .Ta s chng minh trit tiu trn V 1,0 V 1,0 v V 0,1 V 0,1. Tht vy, ta c:

(u iIu, v iIv) = I(u iIu), v iIvC = Iu+ iu, v iIvC= Iu, v+ iu, v i.u, v u, Iv = 0.

Tng t, ta cng c (u+ iIu, v + iIv) = 0. Vy nn 1,1 V .Hn th na, ta c th chng minh c rng nu (V, , ) khng gian Euclide v I

l cu trc hu phc tng thch vi tch v hng ca n th dng( , ) = , i lmt dng Hermit xc nh dng trn (V, I). Ta c mnh sau:

Mnh 1.2.6. Cho (V, , , I) nh trn. Khi qua ng cu chnh tc (V, I) '(V 1,0, i) cho bi v 1

2(v iIv), ta c:

1

2(u, v) = u, vC |V 1,0

Chng minh. Ta c:

u iIu, v iIvC = u, v+ i.u, Iv i.Iu, v+ Iu, Iv = 2(u, v)

Vy ta c iu phi chng minh.

V Th Minh Phng 7


Nhn xt 1.2.7. Gi s {z1, z2, ..., zn} l C- c s ca V 1,0 v gi s zi = 12

(xi iIxi), xi V . Th th {x1 , y1 = Ix1, ..., xn, yn = Ixn} l R- c s ca V v {x1, x2, ..., xn}l C- c s ca (V, I).

Gi s dng Hermit , C trn V 0,1 c ma trn 12(hij) ng vi c s zi th ta c:

i

aizi,j

bjzjC = 12

hijaibj.

Theo Mnh 1.2.6, ta c hij = (xi, xj). Do (, ) l Hermit trn (V, I) nn ta cng c:

(xi, yj) = i (xi, xj) = ihij, (yi, yj) = hij.

Do ( , ) = , i nn = Im( , ) v , = Re( , ). Vy ta c:

(xi, xj) = (yi, yj) = Im(hij),(xi, yj) = (xi, Ixj) = Im(xi, Ixj) = xi, xj = Re(hij),

xi, xj = yi, yj = Re(hij),xi, yj = xi, Ixj = (xi, xj) = Im(hij).

Do :

= i


x1

x

, ...,

xn

x

,

y1

x

, ...,

yn

x

.

, {z1 = x1 + iy1, ..., zn = xn + iyn} l C-c s ca Cn.Trn mi khng gian vc t TxU , ta nh ngha cu trc hu phc I bi:

I : TxU TxU

xi

x

7 yi

x

yi

x

7 xi

x

.

Gi {dx1, ..., dxn, dy1, ..., dyn} l c s i ngu (thc) ca T xU . khi , cu trc huphc cm sinh t I trn T xU cho bi:

I(dxi)

(

xj

x

)= dxi

(I

(

xj

x

))= dxi

(

yj

x

)= 0

I(dxi)

(

yj

x

)= dxi

(I

(

yj

x

))= dxi

( xj

x

)= ij

Do :

I(dxi) = dyi, I(dyi) = dxiTa c mnh sau y:

Mnh 1.3.1. TCU := TU R C c phn tch thnh tng trc tip cc phn thvc t TCU = T

1,0 T 0,1U tha mn I |T 1,0 U = i.Id, I |T 0,1 U = i.Id. Cc phn thvc t T 1,0U, T 0,1U c tm thng ha bi cc nht ct:

zi:=

1

2

(

xi i

yi

)v

zi:=

1

2

(

xi+ i

yi

)Tng t i vi phn th: T CU := T

U C ta cng c phn tch :

T CU = (TU)1,0 (T U)0,1.

Trong (T U)1,0, (T U)0,1 c tm thng ha bi:

dzi = dxi + idyi, dzi = dxi idyi.

Nhn xt 1.3.2. Cho f : U V l nh x nhn. Khi (df)x : TxU : Tf(x)Vc m rng phc bi:

V Th Minh Phng 9


(df)x : TxU C Tf(x)V Cv 7 (df)x(v)

Khi (df)x l C -tuyn tnh.

Mnh 1.3.3. Cho f : U V l nh x chnh hnh gia hai tp m U Cm vV Cn. Khi m rng C -tuyn tnh ca nh x vi phn df : TxU Tf(x)V thamn:

df(T 1,0x U) T 1,0f(x)V v df(T 0,1x U) T 0,1f(x)V

Chng minh. Gi s trn U c ta phc zi = xi + i.yi v trn V c ta phc:

wi = ui + ivi. Gi s thm rng f = (f1, f2, ..., fn) v fi = ai + i.bi Ta c

dfx

(

zi

x

)=

1

2dfx

(

xi

x

i. yi

x

)=

1

2dfx

(

xi

x

) i

2dfx

(

yi

x

)=

1

2

j

ajxi

x

.

uj

f(x)

+1

2

j

bjxi

x

.

vj

f(x)

i2

j

ajyi

x

.

uj

f(x)

i2

bjyi

x

.

vj

f(x)

.

V f l chnh hnh nn ta cajxi

x

=bjyi

x

,ajyi

x

= bjxi

x

. Do

dfx

(

zi

x

)=

1

2

j

(ajxi

x

i.ajyi

x

).

(

uj

f(x)

i. vj

f(x)

).

Vy ta c df(T 1,0x U) T 1,0f(x)V .Chng minh tng t i vi khng nh cn l

nh ngha 1.3.4. Cho U Cn l tp m. Ta nh ngha:p,qU := p(T U)1,0q(T U)0,1v gi AkC(U) v Ap,q(U) tng ng l khng gian cc nht ct ca cc phn th

kCU :=kT CU v p,qU.

Khi ta c phn tch sau y:

V Th Minh Phng 10


kCU =

p+q=k

p,qU, AkC(U) = p+q=k

Ap,q(U).

Ta nh ngha ton t chiup,q : kC(U) p,qU cho bi nu w =

k

p+q=k

dzi1 ... dzip dzj1 ... dzjq thp,q w =

p+q=k

dzi1 ... dzip dzj1 ... dzjp .

nh ngha 1.3.5. Cho d : AkC(U) Ak+1C (U) l m rng C- tuyn tnh ca ton tvi phn ngoi. Khi ta nh ngha:

: Ap,q(U) Ap+1,q(U) v : Ap,q(U) Ap,q+1(U) bi: :=

p+1,q d, := p,q+1 dTa c mnh sau y:

Mnh 1.3.6. Vi cc nh ngha nh trn, ta c:

(i) d = + ,

(ii) 2 = 2

= 0, = ,

(iii) (Lut Leibniz) Ap,q(U), Ar,s(U) :

( ) = + (1)p+q ,( ) = + (1)p+q .

Chng minh. (i) Mnh ny l hin nhin.

(ii) Ta c

0 = d2 = ( + ) ( + ) = 2 + + + 2.

Do ta c ngay 2 = 2

= 0, = .(iii) Ta thy ton t d tha mn:

d( ) = d + (1)p+q d.

V Th Minh Phng 11


Khi vi mi Ap,q(U), Ar,s(U), ta c

( ) =p+r+1,q+s d( )

=p+r+1,q+s

(d + (1)p+q d)=

p+r+1,q+s(d ) + (1)p+q

p+r+1,q+s( d)

=p+r+1,q+s

d + (1)p+q p+r+1,q+s

d

= + (1)p+q .

Vy ( ) = + (1)p+q .ng thc cn li chng minh tng t.

nh ngha 1.3.7. Cho U Cn l tp con m, xt metric Riemannian g trn U . Tani g tng thch vi cu trc hu phc (t nhin) trn U nu:

x U ta c: gx(u, v) = gx(Iu, Iv), u, v TxU

Ta nh ngha dng c bn A1,1U A2(U) bi:

(x)(u, v) := gx(Iu, v).

Metric g khi s cm sinh dng metric Hermit h := g i xc nh dng trnmi khng gian vc t phc (TxU, gx).

V d 1.3.8. Nu g l metric Euclide trn Cn = R2n th:

x1

x

, ...,

xn

x

,

y1

x

, ...,

yn

x

l c s trc giao ca TxU . Ta c dng c bn lin kt vi g c cho bi:

=i

2

ni=1

dzi dzi

Nhn xt 1.3.9. Mt metric bt k g trn U tng thch vi cu trc hu phc c

xc nh duy nht bi ma trn hij(z) := h

(

xi,

xj

). Dng c bn lc c biu

din bi

=i

2

ni=1

hijdzi dzj.

V Th Minh Phng 12


nh ngha 1.3.10. Metric g c gi l mt tip bc hai ti gc ta ti metric

chnh tc nu:

(hij) = Id+O(| z |2).

T nh ngha trn ta d dng suy ra nu g mt tip bc hai ti gc ta ti metric

chnh tc thhijzk

(0) =hijzij

(0) = 0. Ni cch khc l trong khai trin ti im 0 ca

hij thnh chui ly tha th khng xut hin phn t dng aijkzk + a,ijkzk.

Ta c mnh sau y:

Mnh 1.3.11. Cho metric g tng thch vi cu trc hu phc t nhin trn U ,

l dng c bn lin kt. Khi d = 0 khi v ch khi vi mi x U u tn ti ln cnm U

Cn ca x v nh x song chnh hnh f : U f(U ) U tha mn f(0) = xv f g l mt tip bc hai ti gc ta ti metric chnh tc.

Chng minh. Nhn xt rng nu l dng c bn lin kt vi g th f l dng c bn

lin kt vi f g.

Gi s d = 0. C nh p U , ta c th gi s p = 0. Ta vit :

=1

2

hijdzi dzj.

Do ma trn (hij) l Hermit nn ta c th gi s rng (hij(0)) = Id v

hij = ij +k

aijkzk +k

aijkzk +O(|z2|).

Trong aijk =hijzk

(0), aijk =

hijzk

(0). Do d = 0 nn:

hijzk

dzk dzi dzj + hkjzi

dzi dzk dzj = 0.

Vy nn aijk = akji. Tng t ta cng c aijk = a

ikj. Mt khc v hij = hji nn

aijk = ajik. Ta dng php i bin:

j = zj +1

2

i,k

aijk zi zk.

Php i bin ny xc nh cho ta nh x song chnh hnh f . Ta c:

dj = dzj +1

2

i,k

aijk (dzi) zk +1

2

i,k

aijk zi (dzk)

= dzj +i,k

aijk zk dzi.

V Th Minh Phng 13


Tng t ta cng c:

dj = dzj +i,k

aijk zk dzi.

Do nu khng tnh cc thnh phn vi bc ln hn 1 th ta c:

i

2

nj=1

dj dj = i2

(dzj dzj + (n

i,k=1

aijk zk dzi) dzj + dzj (n

i,k=1

aijk zk dzi))

=i

2

(nj=1

dzj dzj +n

i,j=1

(nk=1

aijkzk)dzi dzj +n

i,j=1

(nk=1

ajikzk)dzj dzi

).

Vy nni

2

nj=1

dj dj =

Do f g mt tip bc hai ti metric chnh tc.

Ngc li, gi s tn ti nh x f nh trn. Khi ta c f l dng c bn ng vi

f g. Do f g mt tip bc hai ti gc ta ti metric chnh tc nn o hm

zi,

zica cc h s ca f u trit tiu ti gc. Do d(f )(0) = 0 vy d(p) = 0. Ta c

iu phi chng minh.

V Th Minh Phng 14

Chng 2

Php tnh vi phn trn a tp phc

2.1 Cu trc hu phc trn a tp

nh ngha 2.1.1. Cu trc hu phc trn a tp nhn X l mt ng cu phn th

nhn I : TX TX tha mn I2 = Id.

Vy nu trn X tn ti mt cu trc hu phc th X c s chiu chn. Tuy nhin

khng phi a tp nhn vi s chiu chn no cng chp nhn mt cu trc hu phc.

Ta c mnh sau y:

Mnh 2.1.2. Mi a tp phc X chiu n u chp nhn mt cu trc hu phc t

nhin cm sinh t cu trc phc.

Chng minh. Vi x X, gi (U,) l bn chnh hnh ca X quanh x. Ta gi s rng{zi = xi + iyi}ni=1 l C- c s ca Cn. Khi khng gian tip xc thc TxX c c s l{

x1

x

, ...,

xn

x

,

y1

x

, ...

yn

x

}. Ta nh ngha ng cu I tc ng trn mi TxX

nh sau:

I :TxX TxX

xi

x

7 yi

x

yi

x

7 xi

x

.

Hin nhin I l hon ton xc nh v tha mn I2 = Id. Ta s chng minh I l nhn.

15


Tht vy, gi (U,), (V, ) l cc bn chnh hnh ca X xung quanh x. Khi

ta c cc tm thng ha a phng ca phn th vc t phc pi : TX X cho bi:

|U : pi1(U) (U) Cn,

|V : pi1(V ) (V ) Cn.

Vy, vi mi U, i, i C, ta c:

V I 1U (, 1, 2, ..., n, 1, 2, ..., n)

= V I(

i

xi

1()

+

i

yi

1()

)

= V

(i

yi

1()

i

xi

1()

)=( 1() ,1, ...,n, 1, ..., n

).

Vy I l nhn.

nh ngha 2.1.3. Cho X l a tp hu phc. Ta nh ngha khng gian tip xc phc

TCX := TX R C.

Ta thy rng ng cu I c th m rng C-tuyn tnh trn TCX. Khi ta c mnh sau y:

Mnh 2.1.4. Cho X l a tp hu phc vi cu trc hu phc I. Khi :

(i) Ta c phn tch tng trc tip:

TCX = T1,0X T 0,1X,

Trong T 1,0X,T 0,1X l cc phn th con ca TCX tha mn:

I|T 1,0X = i.Id, I|T 0,1X = i.Id.

(ii) Nu X l a tp phc th T 1,0X ng cu t nhin vi phn th tip xc chnh

hnh TX .

V Th Minh Phng 16


Chng minh. (i) Khng nh (i) l hin nhin.

(ii) Gi s X l a tp phc. Gi i : Ui i(Ui) Cn l bn chnh hnh caX. Khi ta c TCU = T

1,0U T 0,1U , trong T 1,0U v T 0,1U c mc tiu chnh tctng ng l

{

z1, ...,

zn

}v

{

z1, ...,

zn

}. Ta c cc tm thng ha a phng:

i : T1,0Ui Ui Cn,

i

zi|p 7 (p, 1, ..., n).

Ly p UiUj. t ij := i1j . Khi nh x i1j : UiUjCn UiUjCnc cho bi:

i 1j (z, 1, ..., n) = i(

k

j

zk

z

)= i

(k

l

lijzk

j(z)

.

zl

z

)

= i

(l

(k

lijzk

j(z)

).

zl

z

)=

(z,j

j1ijzi

, ...,j

jnijzi

).

Vy phn th vc t T 1,0X c h hm chuyn l: ij(z) = J(ij)(j(z)) nn n ng

cu t nhin vi phn th tip xc chnh hnh TX .

V l do , phn th T 1,0X,T 0,1X c gi l cc phn th tip xc chnh hnh v

phn chnh hnh ca a tp hu phc X.

Nhn xt 2.1.5. Cho E X l phn th vc t phc. Ta nh ngha phn th phclin hp E X l phn th vc t c khng gian ton th l E, cu trc khng gianvc t trn Ex c php cng ging nh trn Ex v php nhn c nh ngha bi:

C Ex Ex(, v) 7 v.

Nu (ij) l h hm chuyn ca E th (ij) s l h hm chuyn ca E cho bi ij(x, v) =

(x, ij(x)v).

Khi ta thy rng hai phn th vc t phc T 1,0X, T 0,1X l lin hp ca nhau.

2.2 Dng vi phn

nh ngha 2.2.1. Cho X l a tp hu phc. Ta nh ngha cc phn th vc t phc:

V Th Minh Phng 17


kCX =

k(TCX) , p,qX = p(T 1,0X)Cq(T 0,1X)v Ak(X), Ap,q(X) tng ng l khng gian cc nht ct ca kCX v p,qX.

Ta d dng chng minh c h qu sau y:

H qu 2.2.2. Ta c cc phn tch:kCX =

p+q=k

p,qX,Ak(X) =

p+q=k

Ap+q(X)

Hn na ta c Ap,q(X) = Aq,p(X) v ngc li Aq,p(X) = Ap,q(X).

Tng t ta cng c th nh ngha cc php chiu:k : A(X) = 2nk=0

Ak(X) Ak(X),p,q : A(X) Ap,q(X)nh ngha 2.2.3. Cho X l a tp hu phc v d : Ak(X) Ak+1(X) l m rngC-tuyn tnh ca ton t vi phn ngoi. Ta nh ngha:

:=p,q d, := p,q+1 d.

T nh ngha, ta d dng thy ton t , cng tha mn lut Leibniz:

( ) = () + (1)p+q (),

( ) = () + (1)p+q ().

Ta c nh mnh sau y:

Mnh 2.2.4. Cho X l a tp hu phc. Khi hai iu kin sau y l tng

ng:

(i) d() = () + (), A(X),

(ii) Trn A1,0(X) ta c 0,2 d() = 0, A1,0(X).Nu X l a tp phc th c hai iu kin trn u tha mn.

Chng minh. (i) = (ii). Gi s d() = () + (). Nu A1,0(X) th:

V Th Minh Phng 18


0,2 d() = 0,2( + ) = 0,2() +0,2() = 0.Vy ta c iu cn chng minh.

(ii) = (i). Ta s chng minh Ap,q(X) th

d Ap+1,q(X)Ap,q+1(X). (2.1)

Tht vy, trc ht ta thy nu A1,0(X) = th d A2,0(X)A1,1(X) dogi thit. Nu A0,1(X) th A1,0(X) = A2,0(X)A1,1(X). Do A0,2(X)A1,1(X).

Ta s chng minh (2.1) bng quy np theo k = p+ q.

- Nu k = 1 th (2.1) ng do chng minh trn.

- Gi s = fi1 ... ip j1 ... jq th ta c

d = (df) i1 ... ip j1 ... jq + f.d(i1 ... ip

j1 ... jq)

= (df) i1 ... ip j1 ... jq + f.d(i1 ... ip

j1 ... jq1)

jq

+ (1)p+q1f.i1 ... ip j1 ... jq1 d

jq .

Theo gi thit quy np th d(i1...ipj1...jq1) Ap+1,q1(X)

Ap,q(X).Do f.d(i1 ... ip j1 ...

jq1)

jq Ap+1,q(X)

Ap,q+1(X). Mtkhc v df A1,0(X)A0,1(X) nn ta c f.i1 ...ip j1 ...jq1 djq Ap+1,q(X)Ap,q+1(X).

Vy (2.1) c chng minh.

Nu X l a tp phc th ta c cc phn t dng dzi1 ... dzip dzj1 ... dzjq lpthnh c s ca Ap,q(X)Ap,q(X) nn c hai iu kin trn u c tha mn.nh ngha 2.2.5. Cu trc hu phc I trn a tp nhn X c gi l kh tch nu

mt trong hai iu kin trong mnh trn c tha mn.

Sau y ta xt mt iu kin khc cho tnh kh tch ca cu trc hu phc I trn

a tp nhn X.

Mnh 2.2.6. Cu trc hu phc I l kh tch nu v ch nu mc Lie ca trng

vc t bo ton T 1,0X , ngha l [T1,0X , T

1,0X ] T 1,0X .

V Th Minh Phng 19


Chng minh. Nhc li rng ta c phn tch TCX = T1,0XT 0,1X, nn mi trng vc t

phc trn TCX c th vit c di dng tng = + i trong T 1,0X , T 0,1X .By gi ly A1,0(X) v v, w l cc nht ct trn T 0,1X. Ta c

(d)(v, w) = v((w)) w((v)) [v, w].

Do A1,0(X) nn (v) = (w) = 0. Vy d A2,0(X)A1,1(X). Do [v, w] T 0,1X , v, w T 0,1X . Vy ta c iu phi chng minh.

T iu kin kh tch trn, ta thu c h qu quan trng sau y.

H qu 2.2.7. Nu I l cu trc hu phc kh tch th ta c

2 = 2

= 0, = .

Ngc li, nu 2

= 0 th I l cu trc hu phc kh tch.

Chng minh. Nhn xt u tin l hin nhin, ta chng minh nhn xt th hai.

Gi s 2

= 0. Ta s s dng Mnh 2.2.6 chng minh tnh kh tch ca I. Ly

v, w T 0,1X , A0,1(X). Ta c

(d)(v, w) = v((w)) w((v)) [v, w].

Mt khc (d)(v, w) = ()(v, w) nn

0 = (2f)(v, w) = v((f)(w) w(f)(v) (f)([v, w]).

V v, w T 1,0X nn ta c

0 = v(df(w)) w(df(v)) (f)([v, w])= (d2f)(v, w) + (df)([v, w]) (f)([v, w]) = (f)([v, w]) do df = f + f.

V { f | f CX} sinh ra A1,0(X) nn ta c [v, w] T 0,1X . Vy I l kh tch.

Ta c nh l sau y v s tng ng gia cu trc hu phc kh tch vi cu trc

phc trn a tp nhn.

nh l 2.2.8 ( nh l Newlander-Nierenberg). Cho I l mt cu trc hu phc kh

tch trn a tp nhn X. Th th tn ti mt cu trc phc trn X tha mn cu trc

hu phc cm sinh trn X trng vi I.

V Th Minh Phng 20

Chng 3

a tp Kahler

3.1 Metric Kahler

Cho X l a tp phc vi cu trc hu phc I.

nh ngha 3.1.1. Mt metric Riemannian g trn X c gi l mt cu trc Hermit

trn X nu metric g tng thch vi cu trc hu phc I, ngha l:

g(I, I) = g(, )

vi mi , l cc trng vc t trn X.

Khi , metric g cm sinh mt dng c bn lin kt A1,1(X) xc nh bi:

(, ) := g(I, ).

D thy c biu din a phng:

=i

2

ni,j=1

hijdzi dzj,

vi mi x X th (hij(x)) l ma trn Hermit xc nh dng.Mt a tp phc X c trang b cu trc Hermit g c gi l mt a tp Hermit.

Lun tn ti cu trc Hermit trn a tp phc X. Tht vy, ly g l metric Riemann

trn X v t h(, ) = g(, ) + g(I, I), vi , l cc trng vc t trn X. D thy

ngay rng h l mt cu trc Hermit trn X.

nh ngha 3.1.2. Mt cu trc Kahler l mt cu trc Hermit g m dng c bn

l ng, ngha l d = 0. Khi , ta ni l dng Kahler v g l metric Kahler.

21


Bi mnh sau y, ta thy rng metric Kahler v mt nh phng, l mt metric

mt tip bc hai ti gc ti metric chnh tc trn Cn.

Mnh 3.1.3. Cho X l a tp phc n chiu, g l metric Kahler trn X. Khi , vi

mi p X th tn ti bn chnh hnh vi h ta (z1, . . . , zn) ti p sao cho ma trn(hij(p)) = h

(

zi,

zj

)trong h ta ny c dng In +O(|z|2).

Chng minh. Ly p X. Gi (U,) l bn chnh hnh ca X quanh p tha mn(p) = 0. V metric g l Kahler nn d((1)) = 0. Theo Mnh 1.3.11 th tn ti

tp m U Cn cha 0 v nh x song chnh hnh f : U f(U) (U) tha mnf(0) = 0 v f ((1)g) l mt tip bc hai ti gc ta ti metric chnh tc. t

V = 1(f(U)), = f1 . Th th (V, ) chnh l bn chnh hnh cn tm.

B sau y cho php ta xc nh metric Kahler t mt dng thc ng, xc nh

dng.

B 3.1.4. Cho l dng thc ng, A1,1(X). Nu xc nh dng, ngha l c biu din a phng

=1

2

ni,j=1

hijdzi dzj

tha mn (hij(x)) l ma trn Hermit xc nh dng vi mi x X, th tn ti metricKahler trn X tha mn l dng c bn lin kt.

Chng minh. Ta xc inh g bi: g(, ) = (, I) vi , l cc trng vc t trn

X. Ta s ch ra g l mt metric Kahler. Tht vy, gi s:

=

i

zi+

i

zi,

=

j

zj+

j

zj.

Ta c cc iu sau:

g(, ) = (, I) = (

i

zi+

i

zi, i

j

zj i

j

zj)

= i

ij(

zi,

zj) + i

ij(

zi,

zj)

=1

2(

hijij +

hjiij).

V Th Minh Phng 22


Tng t g(, ) =1

2(hijij +

hjiij). Do vy g(, ) = g(, ).

g(I, I) = (I, I2) = (I,) = (I, ) = (, I) = g(, ) = g(, ).Vy g tng thch vi cu trc hu phc.

g(, ) = ijhij > 0 do (hij) l ma trn xc nh dng. Do g l dngHermit lin kt vi .

Vy g l metric Kahler.

nh ngha 3.1.5. Ta gi tp cc (1, 1)-dng thc ng v xc nh dng A1,1Xl tp cc dng Kahler.

V d 3.1.6 (Mt s v d v a tp Kahler). 1) Xt khng gian Cn vi metric chnhtc:

d2 =ni=1

dzidzi =n

i,j=1

ijdzidzj.

Khi dng c bn A1,1(Cn) cho bi

=i

2

dzi dzi.

D thy ngay rng d = 0. Vy Cn vi metric chnh tc l a tp Kahler.2) Xt khng gian x nh phc Pn := PnC = (Cn+1 \ {0})/C. Trong C l quan

h tng ng:

z z Cn C | z = z.

Khi c nh x chiu

pi : Cn+1 \ {0} Pn

lm cho Pn c cu trc tp thng. t

Ui = {(z0 : z1 : . . . : zn) | zi 6= 0} Pn.

Khi Ui l m v pi1(Ui) = (Cn \ {0}) \ {zi = 0}.

Xt nh x: i : Ui Cn cho bi:

(z0 : z1 : . . . : zn) 7(z0zi, . . . ,

zi1zi

,zi+1zi

, . . .znzi

).

V Th Minh Phng 23


D thy i l song nh v ta c nh x chuyn ij = i1j : j(UiUj) i(UiUj)cho bi

ij(w1, w2, . . . , wn) =

(w1wi, . . . ,

wi1wi

,wi+1wi

, . . . ,wj1wj

,1

wi,wjwi, . . . ,

wnwi

).

Do vy ij l nh x song chnh hnh. Vy Pn l a tp phc.Metric Fubini-Study trn Pn c cho bi (1, 1) dng lin kt FS c xc nh nh

sau Vi mi:

i : Ui Cn

(z0 : . . . : zn) 7 (z0zi, . . . ,

zizi, . . . ,

znzi

).

Ta nh ngha:

FS|Ui = i :=i

2pi log(

nl=0

zlzi

2). y, i =

i (i), trong i =

i

2pi log(

nk=1

|wk|2 + 1). nh ngha ny l tha ngbi v i|UiUj = j|UiUj . Tht vy, ta c:

log(nl=0

zlzi

2) = log(zjzi

2 nl=0

zlzj

2)= log(

zjzi

2) + log( nl=0

zlzj

2).Do log(

zjzi

2) l hm a iu ha nn log(zjzi

2) = 0 v do i|UiUj = j|UiUj .Vy tn ti dng FS A1,1(Pn).

Ta s ch ra FS l dng thc, ng. Tht vy, v: = = nn ta c i = i,do i l dng thc. Vy FS thc. Mt khc FS hin nhin l ng v i = i = 0

vi mi i. Hn na FS l xc nh dng. Tht vy, ta c:

log(1 +ni=1

|wi|2) =(ni=1

widi

1 +ni=1

|wi|2) =

ni=1

(wi

1 +ni=1

|wi|2dwi)

=

ni=1

widwi

1 +ni=1

|wi|2=

dwi dwi

1 + |wi|2 (

widwi)(

widwi)

(1 + |wi|2)2

=1

(1 + |wi|2)2 hijdwi dwj,

V Th Minh Phng 24


trong

hij = (1 +|wi|2)ij wiwj.

La c, vi mi u Cn (u 6= 0) th:

uthiju =(u, u) + (w,w)(u, u) utwwtu = (u, u) + (w,w) (u,w)(w, u)=(u, u) + (w,w) (w, u)(w, u) = (u, u) + (w,w) |(w, u)|2 > 0.

Vy FS l dng Kahler, do theo B 3.1.4 n cm sinh metric Kahler trn Pn vc gi l metric Fubini - Study.

Tnh cht: (i) Cho pi : Cn+1 \ {0} Pn l php chiu t nhin. Ta c :

pi(FS) =i

2pi log z2.

Tht vy, trn Ui ta c FS = i (i), nn:

pi(FS) = pi(i (i)) = (i pi)(i) =i

2pi log(

nl=0

zlzi

2)=

i

2pi log z2 log |zi|2 = i

2pi log z2.

(ii) Ta c P1FS = 1. Tht vy, ta c:

P1

FS =

C

i

2pi

1

(1 + ||2)2dw dw

=1

pi

R2

1

(1 + (x, y)2)2dx dy = 20

rdr

(1 + r2)2= 1.

3) Xt xuyn phc Cn/, vi = {2ni=1

aiwi , ai Z} v {w1, w2, ..., w2n} l R-c sca Cn. t T = Cn/. Ta c php chiu t nhin

pi : Cn T .

Gi U Cn l mt tp m trong Cn m khng cha hai im no sai khc mt phnt trong . Khi ta c th chng minh c rng U := pi(U) l tp m trong T vnh x

pi|U : U U

V Th Minh Phng 25


l mt ng phi. Vy ta xc nh bn a phng (U,) trn T , trong = (pi|U)1.Khi T vi cu trc phc xc nh nh trn l mt a tp phc n chiu.

Hn na mi xuyn phc Cn/ u c cu trc ca a tp Kahler. Tht vy, vmetric chnh tc g trn Cn l - bt bin nn cm sinh mt metric Kahler trn xuyn.

3.2 c trng ca a tp Kahler

Trong phn ny, chng ti s trnh by cc tnh cht c trng ca a tp Kahler lin

quan n lin thng Levi-Civita v lin thng Chern, mt dng lin thng trn phn

th vc t chnh hnh. Trc tin ta nhc li v lin thng Levi-Civita.

Cho M l a tp nhn. K hiu TM l khng gian cc trng vc t nhn trn M .Nhc li rng mt lin thng D trn M l mt nh x:

D : TM TM TM

(, ) 7 D(, ) := D

sao cho D l R-song tuyn tnh v C(M)-tuyn tnh theo , ng thi tha mn lutLeibniz sau:

D(f) = fD + (f)D

vi mi f C(M).Cho (M, g) l a tp Riemannian. Lin thng D trn M c gi l tng thch vi

metric g nu

(g(, )) = g(D, ) + g(,D).

Ta c nh l c bn ca hnh hc Riemannian sau:

nh l 3.2.1. Cho (M, g) l mt a tp Riemannian. Khi tn tai duy nht lin

thng D m ta gi l lin thng Levi-Civita trn (M, g) tha mn cc tnh cht sau y:

(i) D tng thch vi metric g.

(ii) D l khng xon, ngha l D tha mn D D = [, ] vi mi trng vc tnhn , trn M .

V Th Minh Phng 26


Nhn xt 3.2.2. K hiu A1(M,TM) l khng gian cc nht ct nhn ca phn thvc t (nhn) T M

TM . Khi ta c th xc nh lin thng D nh nh x R -tuyn

tnh :

D : TM A1(M,TM)

v tha mn:

D(f) = fD() + df .

3.2.1 Phn th vc t Hermit v lin thng Chern

Cho E X l phn th vc t phc trn a tp thc X.

nh ngha 3.2.3. Mt metric Hermit h trn E X l cho trn mi th Ex mt tchHermit , x tha mn vi mi tp m U X v vi mi , A(U,E) l nht ct aphng trn U th nh x , : U C cho bi , (x) := (x), (x)x l mt hmnhn. Mt phn th vc t E c trang b metric Hermit c gi l mt phn th vc

t Hermit.

Nu (U, ) l mt tm thng ha a phng : E|U U Ck th ta c mtmc tiu a phng (e1, e2, ..., ek) ca E trn U , khi ta c th ng nht mi nht

ct ca E trn U vi k hm (1, 2, ..., k), trong i : U C. Ta xc nh matrn (hij) : U Gl(k,C), vi hij C(U) bi:

hij(x) := ei, ej(x).

Khi vi mi , A0(U,E), ta vit = i

iei, =j

jej v ta c:

, = i

iei,j

jej =i,j

ijei, ej =i,j

ijhij.

nh ngha 3.2.4. Ap(X,E) = A(X,pT XCE) c gi l l khng gian cc dngvi phn bc p trn X vi h s trong E.

Nhn xt 3.2.5. Mi nht ct a phng Ap(U,E) vi U m trong X c th vitdi dng =

i

i i. Trong i Ap(U) l p-dng vi phn , i A0(U,E) lnht ct ca E trn U

Sau y ta xt n lin thng trn phn th vc t phc.

V Th Minh Phng 27


nh ngha 3.2.6. Cho E X l mt phn th vc t phc. Khi ta nh nghalin thng trn E l mt nh x C-tuyn tnh:

: A0(X,E) A1(X,E)

tha mn lut Leibniz

(f) = df + f

vi mi f C(U), A0(U,E), y U m trong X.

Khi nu (U, ) l tm thng ha a phng ca E. Trong mc tiu a phng

lin kt (e1, e2, ..., ek) ta c th biu din di dng ta :

(1, 2, ..., k) = (d1, d2, ..., dk) + (1, 2, ..., k).A.

Trong A l ma trn kk vi h s l cc 1-dng trn U . Nh vy v mt a phngta c th vit = d+ A trong A l ma trn cc 1-dng.

Gi s h l metric Hermit trn phn th vc t E. Khi ta c th m rng h ti

nh x (m ta vn k hiu l h) cho bi:

h : Ap(X,E)Aq(X,E) Ap+q(X)1 1, 2 2 7 1 2.1, 2

nh ngha 3.2.7. Lin thng trn E c gi l tng thch vi metric Hermit hnu:

d, = , + ,vi mi , A(X,E)

Gi s 5 c biu din a phng l = d+ A. Ta c:

dei, ej = ei, ej+ ei,ej

Do vy

dhij =k

Aik ek, ej+ ei,l

Ajl el

=k

Aikhkj +l

Ajlhil

= (A.h)ij + (h.At)ij.

V Th Minh Phng 28


Tip chng ta s i nh ngha khi nim lin thng Chern v ch ra s tn ti duy

nht ca lin thng Chern trn phn th vc t Hermit chnh hnh. Gi s E X lphn th vc t chnh hnh trn a tp phc X. Nhc li rng:

A(X) =r=1

Ar(X),

trong Ar(X) = p+q=r

Ap,q(X). Do ta cng c phn tch:

A(X,E) =r

Ar(X,E), Ar(X,E) =p+q=r

Ap,q(X,E).

Ap,q(X,E) = Ap,q(X)A0(X,E). T nhn xt trn, ta thy mi lin thng trn E u c th vit duy nht di dng = 1,0 +0,1, :

1,0 : A0(X,E) A1,0(X,E),0,1 : A0(X,E) A0,1(X,E).

Nhn xt rng, trn phn th vc t chnh hnh E X tn ti ton t E :A(X,E) A0,1(X,E) c nh ngha a phng bi: vi mi tm thng ha aphng U,), gi s (e1, e2, ..., ek) l mc tiu a phng ca E trn U v (1, 2, ..., k)

l ta a phng ca nht ct th:

E = U|U := (1, 2, ..., k).

Ton t E l hon ton xc nh qua b sau y.

B 3.2.8. Gi s (U,), (V, ) l cc tm thng ha a phng ca E tha mn

U V 6= . Khi vi mi A0(X,E). Ta c:

U|UV = V |UV .

Chng minh. Gi (e1, e2, ..., ek) v (e1, e

2, ..., e

k) l cc mc tiu a phng ca E

lin kt vi (U,) v (V, ) tng ng. Gi s g = (gij) l ma trn chuyn mc tiu

t (e1, e

2, ..., e

k) sang (e1, e2, ..., ek), vi gij l cc hm chnh hnh trn U V . Gi

(1, 2, ..., k) v (,1,

2, ...,

k) tng ng l ta ca i vi hai mc tiu trn.

Khi ta c k =

i i.gki. Do gij l cc hm chnh hnh nn ta c:

k =i

(i.gki) =

i

(i.gki +

i.gki) =

i

i.gki.

V Th Minh Phng 29


Ta cn chng minh U|UV = V |UV , hay

k k.ek =

i i.ek. Tht vy, ta c:

k

kek =k

i

i gki ek =

i

i

(k

gki ek

)=

i

i.ek.

Vy ta c iu phi chng minh.

Ta c nh l sau y.

nh l 3.2.9. Tn ti duy nht lin thng trn E tha mn cc iu kin sau y:

(i) tng thch vi metric Hermit h.

(ii) 0,1 = E

Lin thng duy nht xc nh nh trn gi l lin thng Chern ca phn th E.

Chng minh. Trc tin ta ch ra s tn ti ca lin thng cn dng.

- -Xt trng hp E = X Cn l phn th vc t tm thng, khi ta xc nhlin thng bi = d+ A, trong A = h.h1.

- -Trng hp E l phn th vc t bt k. Xt vi mi tm thng ha a phng

(U, ) ta nh ngha lin thng tng t nh trng hp trn. Gi (f) lphn hoch n v ng vi ph (U). Ta nh ngha lin thng trn E nh sau:

=

f..

D dng kim tra c tha mn lut Leibniz.

kim tra tnh duy nht, ta vit = 1,0 +0,1. Khi tng thch vi metrich nu v ch nu:

d(h(, )) = h(, ) + h(,).So snh phn dng (1, 0) ca hai v ta c

h(, ) = h(1,0, ) + h(,0,1).

V 1,0 = nn ta ch(, ) = h(1,0, ) + h(, ).

V Th Minh Phng 30


Gi (e1, e2, ..., ek) l mc tiu chnh hnh ton cc trn E. Th th ei = 0.Vy ta c:

h(1,0ei, ej) = (h(ei, ej)),

hay h(1,0ei, ej) = hij. T biu thc ny ta thy lin thng Chern trn E nu tnti l duy nht. Tht vy, gi s c biu din a phng l = d + A, khi (A.h)ij = (h)ij, do vy A = h.h

1.

3.2.2 Cc tnh cht c trng ca a tp Kahler

Cho X l a tp phc vi I l cu trc hu phc t nhin trn TX. Nhc li rng khi

ta c TCX = T1,0X

T 0,1X. Trong phn th T 1,0X ng cu t nhin vi phn

th vc t chnh hnh TX . Gi s g l cu trc Hermit trn X, ngha l mt metricRiemannian tng thch vi cu trc hu phc I. Khi g cm sinh dng c bn

(u, v) = g(Iu, v), A1,1(X)

v cm sinh metric Hermit gC trn phn th vc t TCX. Ta ch rng gC|T 1,0X =1

2(g i) v cc phn th vc t (TX, I) v (T 1,0X, i) c ng nht vi nhau qua

ng cu :

R :TX T 1,0Xu 7 1

2(u iIu).

Do ng cu R c tnh cht I R = R i nn vi mi trng vc t thc trn X talun ng nht i. vi I trn TX. V T 1,0X l phn th vc t chnh hnh nn trn

(T 1,0X, gC) tn ti duy nht lin thng Chern v trn (TX, g) tn ti duy nht linthng Levi-Civita D. nh l sau y s ch ra mi quan h ca chng trong trng hp

X l a tp Kahler.

nh l 3.2.10. Cho X l a tp phc, g l cu trc Hermit trn X vi cu trc hu

phc I. Khi cc mnh sau l tng ng:

(i) g l metric Kahler.

(ii) Cu trc hu phc I l phng i vi lin thng Levi-Civita D, ngha l D(I) =

I(D) vi mi l trng vc t trn X.

V Th Minh Phng 31


(iii) Qua ng cu R, lin thng Chern v lin thng Levi-Civita D l trng nhau.

Chng minh. Ta chng minh theo cc bc sau.

(iii) = (ii). V qua ng cu R lin thng Chern v lin thng Levi-Civita Dtrng nhau nn D c xc nh bi:

D() = (R),

vi l trng vc t trn TX. y ta lun ng nht i. vi I trn TX. V lC-tuyn tnh nn ta c:

(iR) = i.(R),hay

(RI) = i.(R).Do D(I) = ID(X). Vy ta c (ii).

(ii) = (i). Ta c g(u, v) = (u, Iv). V D l lin thng Levi-Civita nn ta cdg(, ) = g(D, ) + g(,D) vi mi , A0(TX). Do vy:

d((, ) = dg(I, ) = g(DI, ) + g(I,D) = g(ID, ) + (,D)

= (D, ) + (,D).

Do vi mi A0(TX) ta c:

((, )) = (D, ) + (,D). (3.1)

Mt khc, ta c:

d(, , ) = ((, )) ((, )) + ((, )) ([, ], ) + (, [, ]) + ([, ], )

= ((, )) ((, )) + (()) (DD, ) + (,D D) + (D D, ).

T y v t (3.1) ta c ngay d(, , ) = 0, , A0(TX). Vy l dng ng,v do g l metric Kahler.

(i) = (iii). Gi s g l metric Kahler. Gi l lin thng Chern trn T 1,0X. Tanh ngha lin thng D trn TX bi:

D = (R).

V Th Minh Phng 32


Ta s chng minh rng D chnh l lin thng Levi-Civita trn TX.

Tht vy, v l lin thng Chern nn n tng thch vi metric Hermit gC|T 1,0X =12(g i) nn

dgC(R,R) = gC(R,R) + gC(R,R).

Do

dg(, ) = g(D, ) + g(,D).

Vy D tng thch vi metric Riemannian g.

Gi s h = (hij) l ma trn biu din a phng cho metric Hermit gC trn T1,0X

v = d + A, trong A = (aij) l biu din a phng ca lin thng Chern. Theochng minh nh l 3.2.9 th ta c A = h.h1. Do :

aij =k

hik hij,

vi (hij) l ma trn nghch o ca ma trn h. Ta s chng minh D khng xon, ngha

l D D = [, ]. Nhc li rng{

z1, ...,

zn

}l mc tiu a phng ca T 1,0X. Khi :{

x1, ...,

xn,

y1, ...,

yn

}l mc tiu a phng ca TX. Ta c:

D

xi=

zi= (d+ A)

(

zi

)=k

aik zk

.

Vy nn:(D

xi

)(

xj

)=k

aik

(

xj

).

zk

=k

aik

(

zi+

zi

).

zk

=k

aik

(

zj

).

zk(v A l ma trn (1,0)-dng).

V Th Minh Phng 33


M aij =l

hil.hlj, nn

(D

xi

)(

xj

)=k

(l

hilzj

.hlk

)

zk.

Do ta cng c: (D

xj

)(

xi

)=k

(l

hjlzi

.hlk

)

zk.

n y, vi ch rng g l metric Kahler nn dng c bn =i

2hijdzi dzj l ng

v ch thm rng ma trn (hij) l Hermit nn ta c vi mi i, j, k th:

hijzk

=hkjzi

,hijzk

=hikzj

,hijzk

=hjizk

. (3.2)

Vy (D

xj

)(

xi

)=

(D

xi

)(

xj

).

Tng t nh vy, p dng (3.2) ta cng chng minh c cc ng thc sau:(D

yj

)(

yi

)=

(D

yi

)(

yj

),

(D

xj

)(

yi

)=

(D

yi

)(

xj

),(

D

yj

)(

xi

)=

(D

xi

)(

yj

).

T y d dng thy rng D l khng xon. Vy D l lin thng Levi-Civita. nh l

c chng minh.

V Th Minh Phng 34

Ti liu tham kho

[1] Raymond O.Wells, Differential Analysis on Complex Manifolds, Springer-Verlag,

2000.

[2] Daniel Huybrechts, Complex Geometry, An Introdution, Springer-Verlag, Berlin,

2005.

[3] Claire Voisin, Hodge Theory and Complex Algebraic Geometry, I, Cambridge

Stud.Adv.Math.76, 2002.

35

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