Let (M, J, g) be a compact K/ihler manifold of constant scalar curvature.Then the K~hler class [w] has an open neighborhood in HI'I(M, R) consisting of classes which are represented by KKhler forms of extremal K~dder metrics; a class in this neighborhood is represented by the K~ihler form of a metric of constant scalar curvature iff the Futaki invariant of the class vanishes. If, moreover, the derivative of the Futaki invariant at [w] is "nondegenerate," every small deformation of the complex manifold (M, J) also carries Kiihler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal K~hler metrics on certain compact complex surfaces.

We give examples of complete locally asymptotically flat Riemannian 4-manifolds with zero scalar curvature and negative mass. The generalized positive action conjecture of Hawking and Pope [5] is therefore false.The positive action theorem, proved by Schoen and Yau [9], states that any complete asymptotically flat Riemannian 4-manifold (M, g) with scalar curvature R = 0 satisfies lim $(g jk . k -g kktj )*dS j ^Q r-*ac S r with equality iff g is flat; here the integrand is to be computed in an asymptotic coordinate system in which the metric is of the form ~2 ana S r denotes the Euclidean sphere of radius r. For simplicity we will call the left-hand side of the above inequality the mass of (M, g), since this expression is the analog of the ADM mass of an asymptotically flat 3-manifold. The generalized positive action conjecture of Hawking and Pope [5] asserts that the mass is also non-negative for all locally asymptotically flat Riemannian 4-manifolds with scalar curvature R = 0, and that, moreover, the mass vanishes iff the manifold is Ricci flat with self-dual Weyl curvature.Unfortunately, as we will see, this plausible-sounding extension does not hold water. We will produce an infinite number of complete locally asymptotically flat Riemannian 4-manifolds with R = 0 for which the mass is negative. These metrics are Kahler and live on the total spaces of complex line-bundles over S 2 -CP : for which the first Chern class satisfies c 1 < -2. They have isometry group (7(2), and, by virtue of being Kahler with R -0, have anti-self-dual Weyl curvature; cf. [2,7,8].

It is shown that there are infinitely many compact orientable smooth 4-manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality 2χ > 3|τ |. The examples in question arise as non-minimal complex algebraic surfaces of general type, and the method of proof stems from Seiberg-Witten theory. * Supported in part by NSF grant DMS-9505744.

We prove that any compact complex surface with c 1 > 0 c_1>0 admits an Einstein metric which is conformally related to a Kähler metric. The key new ingredient is the existence of such a metric on the blow-up C P 2 # 2 C P 2 ¯ \mathbb {CP}_2\# 2\overline {\mathbb {CP}_2} of the complex projective plane at two distinct points.