1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

64 2. KAHLER-RICCI FLOW

EXERCISE 2.11 ( J-invariance of curvature). Show that for a Kahler man-
ifold Rm (X, Y) is J-invariant, i.e., Rm (X, Y) JZ = J (Rm (X, Y) Z).

SOLUTION. We compute

Rm (X, Y) JZ = \7x\7yJZ - \7y\7xJZ - \J[X,Y]JZ

= \Jx (J\JyZ) - \ly (J\JxZ) - J (\J[X,YJZ)

= J (\7x\7yZ) - J (\7y\7xZ) - J (\J[X,YJZ)

= J (Rm (X, Y) Z).

The components of Rec are defined similarly:

Rec ( a~a, a~f3) ~ Rai3'

Rec (a~a' a~/3) ~ Raf3,

etc. Tracing (2.4), we see that components of the Ricci tensor Rai3 = R~i3a

are given by

LEMMA 2.12 (Kahler Re).

(2.6)

a2

azaaz/3 log det (g'YJ).

It is easy to see that Rai3 = Rf3a and Raf3 = Rai3 = 0.

. EXERCISE 2.13. Show that for a Kahler manifold Re is J-invariant, i.e.,

Re (JX, JY) =Re (X, Y).

The Ricci form p is the 2-form associated to Re,

1

p ( X, Y) ~ "2 Re ( J X, Y) ,

which is also a real (1, 1)-form.

EXERCISE 2.14. Show that if g is a Kahler metric, then the Ricci form

p is a closed 2-form.

HINT: See Proposition 2.4 7 on p. 7 4 of [27].

The real de Rham cohomology class [ 2 ~p] ~ c 1 (M) is the first Chern

class of M. It is a beautiful fact that [ 2 ~p] only depends on the complex

structure of M.

The Ricci form, which is a real (1, 1)-form, is i~ holomorphic coordinates:

P = HRai3dza /\ dz!3.

From (2.10) below and r~/3 = ria we have

a a

a za RM = a z/3 Ra;y'