1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

60 2. KAHLER-RICCI FLOW

EXERCISE 2.3. Given a (p, q)-form

n 'I = 'iil n. ·· ·tpJl. ~ ... ~ Jq di^1 /\ • • • /\ dziP /\ dz3^1 /\ • • • /\ dzJq '

show that its complex conjugate is given in local coordinates by

ii::::: 'I • (-l)pq 'lt1 n. ···tpJl.. ··•Jq. dzj^1 /\ • • • /\ dzjq /\ dz?;^1 /\ • • · /\ dz?;P E n,q,p (M).

The exterior differentiation operator d maps Ap,qM into AP+l,qM EB

Ap,q+l M. Corresponding to this decomposition of the image of d, we have

where

d =:=a+ a,

[): Ap,qM ---t AP+l,qM,

8 : Ap,q M ---t Ap,q+^1 M.

We extend the Riemannian metric g complex linearly to define

gee : TccMp x TccMp ---t C.

Similarly

V' cc : TccM x C^00 (TccM) ---t C^00 (TccM)

is the complex linear extension of V' : TM x C^00 (TM) ---t C^00 (TM) with

the convention that V' x Y =:= V' ( X, Y) and (V' cc) x Y =:= V' cc ( X, Y). The

complex linear extension of Rm is denoted by Rmcc.
Let

(

[). [))

g73a =:= gee [)z/3' [)za ·

Since gee (V, W) = gee (V, W) = gee (W, V) , these coefficients satisfy the
Hermitian condition:

Similarly, we define

ga/3 =:= gee ([)~a' [)~/3) '

We claim ga/3 = ga13 ='0. Indeed, if X, YE T^1 ,o M, then

gee (X, Y) =gee (JX, JY) =gee (Rx, HY) =-gee (X, Y),

which implies gee (X, Y) = 0. Similarly, if X, Y E ro,i M, then we also have
gee (X, Y) = 0. Thus, in local holomorphic coordinates, the Kahler metric
takes the form

gee = ga13 ( dza ® dz/3 + dz/3 ® dza).

EXERCISE 2.4. Show that if a= Haa13dza /\ dz/3 is a (1, 1)-form, then
a is real if and only if