On the Scalar Curvature of Complex Surfaces

Let (M, J) be a minimal compact complex surface of Kähler type. It is shown that the smooth 4-manifold M admits a Riemannian metric of positive scalar curvature iff (M,J) admits a Kähler metric of positive scalar curvature. This extends previous results of Witten and Kronheimer. A complex surface is a pair (M,J) consisting of a smooth compact 4-manifold M and a complex structure J on M ; the latter means an almost-complex structure tensor J : TM → TM , J = −1, which is locally isomorphic to the usual constant-coefficient almost-complex structure on R = C. Such a complex surface is called minimal if it contains no embedded copy C of S such that J(TC) = TC and such that C · C = −1 in homology; this is equivalent to saying that (M,J) cannot be obtained from another complex surface (M̌, J̌) by the procedure of “blowing up a point.” A Riemannian metric g on M is said to be is said to be Kähler with respect to J if g is J-invariant and J is parallel with respect to the metric connection of g. If such metrics actually exist, (M,J) is then said to be of Kähler type; by a deep result [1] of Kodaira, Todorov, and Siu, this holds iff b1(M) is even. The purpose of the present note is to prove the following: Theorem 1 Let (M,J) be a minimal complex surface of Kähler type. Then the following are equivalent: (a) M admits a Riemannian metric of positive scalar curvature; (b) (M,J) admits a Kähler metric of positive scalar curvature; (c) (M,J) is either a ruled surface or CP2. Supported in part by NSF grant DMS-9003263.