1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. EXISTENCE OF KAHLER-EINSTEIN METRICS 71

where w = Hwa~dza /\ dz/3 and wa~ = Wf3a·

PROOF. This is a standard result in the theory of Kahler manifolds; see

the book by Zheng [383] for example. D

REMARK 2.27. More generally, we have the following. Let b be a (p, q)-

form, where p, q > 0. If b is d-closed and either d-, 8-, or 8-exact, then

there exists a (p - 1, q - 1)-form <p such that 881.p = b. When p = q and bis

real, we may take H<p to be real. See Lemma 9.1 on p. 221 of [383] for

example.

A fundamental problem in Kahler geometry is the Calabi conjecture,

which was solved by Yau [378, 379] and says the following. (Also see

Theorem 2.30 below.)

THEOREM 2.28 (Calabi conjecture: prescribing the Ricci form in a

Kahler class). Let (Mn, go) be a closed Kahler manifold with Kahler class

[wo]. For any closed real (1, l)-form w E c1 (M), there exists a Kahler metric

g with [w] = [wo] such that its Ricci form is the prescribed form:

(2.25) p = 27rw.

Since Po = HRc (go)a~ dza /\ dz/3 is a real (1, 1)-form in the same
cohomology class 27rc 1 (M) as 2nw = 27rHwa~dza /\ dz/3, by Lemma 2.26
there exists a real-valued function f such that

Re (go) a~ - 27rwa~ = 8a813f.

Therefore equation (2.25), which in local coordinates is Ra~ = 27rwa~' is
equivalent to

8a_813f =Re (go)a~ - Ra~

(^82) ( )^82

= - 8 za 8 z/3 log det (go) 7 J + 8 za 8 z/3 log det (9 7 5).

Since M is closed, this implies

det (9 7 5)

log ( ) = f + log C

det (go) 7 J

for some constant C > 0. Since the real (1, 1)-forms wo and w are in the

same cohomology class, using Lemma 2.26 again, we see that there exists a
real-valued function <p such that

97J = (go)'YJ + 87801.p.

Thus we may rewrite (2.25) as a complex Monge-Ampere equation:

det ( (go) 7 J + 87 851.p)

-~------- eel

det ( (go) 7 J) -.