Some recent developments in Kähler geometry and exceptional

99910_6PL020.pdf උඈർ ඇඍ ඈඇ඀ ඈൿ ൺඍඁ ංආඈඇ ඈඇൺඅൽඌඈඇ 78

X

n X !0 (1;1) za ! 0=iX a;bg abdzad z b; (gab) !0 X !0 [!0]2H2(X;) ! =!0+i@ @ gab g ab+@2 @z a@ z b: (1;1) K 1X= nTX =i@ @L;L L= g ab+@2 @z a@ z b : S X  =!= 1;10 c1(X) S= c1(X);[!] c1(X) [!] S !=Lv! v Lv v= 0 v @! @t =!: @! @t =i@ @S; =1= 0 =1 = 1 X

E (X;!0)

c1(E) = 0 E E E SO(E) c1(S)<0 X L (X;L) (X;L)

X :X! L!X

L 

 (X;L) 1(1) XLj1(1)=Lr r  X X X0=1(0) X0=1(0) L

0 

(X0;L0) k! 1 H0(X0;Lk0) (X) (X;L) (X)0 X

X  X

X c1(L) (X;L) X0 c1(L0)X0 S1

HX0

(X) =Z X

0(S^S)H d;

S X0^S S

R X;L=M kH

0(X;Lk):

^K kXk Lp H (X;L)  >0 X (X)kXk: (X;L) (X;L)

Pn d

P L

P(f) =Z

@P fdAZ P fd; A LP f Z P f d= 0;Z P f x id= 0 xi f f=frg frg QPL f P f2QPL LP(f)>0 fP L

P(f)>0

 >0 f2QPL L

P(f)Z

P jfjd: ^K (X;L) P X X0 X 2 1 (!s) = (1s)+s!s;  (1;1) !t X ti! 1 t0i (X;!t0i) Z (X;!t) t! 1 Z=X X X Z

X

Z X nX =! (2)n(n+1)n X=n =g g K Z p2Z p Zp Z  p(f) =r!0 Brjfj r;

Br r Z p

T=()rN WN W T r= 1  W W 0 4 (X) (X)>1  (X) =k!1k(X)  k=D(X;D): kDP iDi siH0 k=H0(X;KkX) kH0 k
1  [!0] H

H=f:!0+i@

@ >0g: H kk2=Z X ()2d; d !n/n! H (X;!0) H S1 (1;1) Ω(0;1)S1XΩn+1= 0 H 1 H F F=Z X ()(S^S)d;

S ^S

H H C1;1 H V ! !0 V !n0=!n 0 !0

F() =Z

X V Vd0+J0(); J 0=Z X ()0^!n1: FC1;1 H F H H L1

L2

E

1

L1 E1 H (M;g) mp2M p p (M;g) G

SO(m)

G=SO(m)

m G=U(m/2)G=SU(m/2) m 4G=Sp(m/4)G=Sp(m/4):Sp(1) m=m/2 m=m/4 Sp(n):Sp(1) Sp(n) Sp(1) 78
G2SO(7) (7)SO(8) 7 ;;

V m m92< p < m2

pV GL(V) GL(V) m= 7p= 3 V 23V V v7!iv()^iv()^: 7V  3 3 GL+(V) GL+(V) 3 G2 3 g

V

g jj2= 7 g   g   W=V t 3 4 Ω =^dt+24(W): ΩGL+(W) (7) SO(7) A GL+(W)Ω 4W A 6421 = 43 A

G

GSO(m) G G G G2 7  d= 0d= 0 (7) 8 ΩdΩ = 0 G2 G2 G2 T/T G2 3

SO(4)G2SU(3)G2 4

!1;!2;!3

22+4 3t1;t2;t3

3 =dt1dt2dt3X! idti 7 43 42+ SO(4) Q !iQ

3 Q3QT3

G2 T7/ T34/1

Q 4/1

T3Q Q Q   G2 3 SU(3) (1;1)!  za=xa+iya !=Xdx adya; =dz1dz2dz3: !^dt+(); 3 t

Z SU(3) Z

! 3 

Z Z

= 0 G2 ZZS1

Z

W

DW D K3

W=3D CD ~DD ~WW

C ~D

D Z ~D~W

~DS1 ~D=D ZS1 G2 ~

DS1S1:

Z1;Z2 ~DiS1S1 L 7M S1 S1 M L K3~Di K3

M

G2 M L

G2  0 L! 1 G2 d= 0;d= 0 G2 V  ()27V   =1 3 ^(): M7 c2H3(M;)Pc 3 c Pc

V() =Z

M (): Pc =d V=1 3 Z M ^d()); d= 0 V

M V

Pc G2 c

0 cH3(M;) Pc

Pc0 0

MT G2 M

0 7![]

TH3(M;) @ @t = ∆; ∆ g  @g @t =2+Q();

Q() =d

 dV dt =kdk2; G2 Pc G2 7

N/

3 G2 0

MH1(M;) =

0

G2 S7   G2 G2 G2 K 1Z K1Z G2 G2

MH3(M;)

b3 p1(M)2 H

4(M;) C

S3S4] S3S4:::] S3S4;

CS3S4 S

3 S4

G2 G2 (7)/G2 G

2

M!(7)/G2=S7

2/48  (W)3(W) W

M (7) G2

= 24  ^ G2 1 G2 G2 (Mi;gi) pi2Mi (M1;p1) m  L! 1 m Mi G2

K3

X G2(M;) X :M!BB  LB L L M=YS1;B= S

2S1 Y S2

B0BnL 4X B0 f:B0!+

3(H2(X;));

H2(X) (3;19) G2 f H2(X;)

B

3 :

M!Y YS1G2

M  !0  G2 Y G2;(7) M G 2G 2 2 G

E!M2G

E

G

(7);G2;SU(3) SU(3) (7) G

2

SU(3)

D (7) S4 S G2 t

G2 t G2

E t0 Att < t0 t!t0 P t P P P 1 G2 K G2 2 Cnn3 2 G2 G2 G2 ғ 2 4 G2 G2 G2 K3

8 (7)

7G2 G2 C3 2 K G2 K n (2;C) 2 ()

G2

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