Page 258 - Curvature and Homology

240 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS

(a) v is generated by X and JX. Then, a(X,Z) = a(JX,Z) = 0 for any
Z E {X, Jx}~-the orthogonal complement of the space generated by X and JX.
(b) v is generated by X, JX,Y, JY where X and Y have the property that
a(X,Z) = a(Y,Z) = 0 for any Z E {x, Y] : Put Y = a JX + b W where W is
a vector defined by the condition that {x, Jx, W, JW} is an orthonormal set.
The only non-vanishing components of a on v are given by a(X, JX), a(W, JW),
a(X, W) = a( JX, JW). Therefore, when Z E PA, a(X,Z) = a( JX,Z) = a(W,Z) =
a(JW,Z) = 0.

C. The Ricci curvature of a A-holomorphically pinched Kaehler manifold
1. The Ricci curvature of a bpinched manifold is clearly positive. Show that
the Ricci chrvature of a A-holomorphically pinched Kaehler manifold is positive
for h 2 4.
In the notation of (1.10.10)

Choose an orthonormal basis of the form {X, JX) u {Xi, JXi} (i = 2, -, n)
and apply (A. 5).

D. The second betti number of a compact &pinched Kaehler manifold [2]
1. Prove that for a 4-dimensional compact Kaehler manifold M of strictly
positive curvature, bdM) = 1.
In the first place, by theorem 6.2.1 a harmonic 2-form a is of bidegree. (1,l).
By cor. 5.7.3, a = r52 + p, Y E R where 52 is the fundamental 2-form of M
and p is an effective form (of bidegree (1,l)). Since a basis may be chosen so
that the only non-vanishing components of p are of the form pi,,, then, by
(3.2.10),

Applying (A. l(d)) we obtain

Finally, since Kij + Kij, > 0 and p is effective, it must vanish.

2. If M is A-holomorphically pinched with A > 4, then b,(M) = 1.
Hint: Apply AS.
3. Show that (D.2) gives the best possible result. (It has recently been shown
that a 4-dimensional compact Kaehler manifold of strictly positive curvature