- INTRODUCTION TO KAHLER MANIFOLDS 61
Given a ]-invariant symmetric 2-tensor b, we define a ]-invariant 2-form
,8 by
(3(X,Y) ~b(JX,Y).(We check that (3 (Y, X) = b (JY, X) = b (-Y, JX) = -(3 (X, Y) .) We call (3
the associated 2-form to the symmetric 2-tensor b. The Kahler form w,on a Riemannian manifold ( M, J, g) with an almost complex structure and
whose metric is Hermitian, is defined to be the 2-form associated tog:w(X,Y) ~g(JX,Y),which is a real (1, 1)-form.EXERCISE 2.5. Show that if g is a Kahler metric, then w is a closed
2-form. In fact, w is parallel.HINT: See [27], Proposition 2.29 on p. 70.SOLUTION: We compute(V' xw) (Y, Z) = X ( w (Y, Z)) - w (V' x Y, Z) - w (Y, V' x Z)= X (g ( JY, Z)) - g ( J (V' x Y) , Z) - g ( JY, V' x Z)
= (V'xg) (JY,Z) = 0,
where we used the definition of w, J (V' x Y) = V' x ( JY) , and g is parallel.
On a Kahler manifold (M, J, g), in local holomorphic coordinates, the
Kahler form is
n
w = H L 9a'i]dza /\ dz/3.
a,(3=1REMARK 2.6. In our convention, dza /\dz/3 ( 8 ~"1, 8 ~ 8 ) = ~5~5f, Note that
for the standard Euclidean dx^2 + dy^2 metric on ccn, we have 9a'i] = ~5af3 and
9<C = ~ I:~=l ( dza ® dza + dza ® dza).
Since g is Kahler, we have the Kahler identities:(2.2)a a
[)z'Y9a'i] = [)za9'Y°iJ'which are equivalent to dw = 0. The real cohomology class
[w] E H^1 •^1 (M; JR.) ~ H^2 (M; JR.)is called the Kahler class of w.
EXERCISE 2. 7 (Characterization of Kahler condition). If ( M, J, g) is a
triple consisting of a Riemannian manifold, an almost complex structure,
and a Hermitian metric, then dw = 0 if and only if V' J = 0. That is, g is