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Let (M, J) be a compact complex 2-manifold which which admits a Kähler metric for which the integral of the scalar curvature is non-negative. Also suppose that M does not admit a Ricci-flat Kähler metric. Then if M is blown up at sufficiently many points, the resulting complex surface (˜ M , ˜ J) admits Kähler metrics with scalar curvature identically equal… (More)
We show that every closed symplectic four-dimensional manifold (M, ω) admits an almost Kähler metric of negative scalar curvature compatible with ω.