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Let X be a 4-manifold with contact boundary. We prove that the monopoleinvariants of X introduced by Kronheimer and Mrowka vanish under the followingassumptions: (i) a connected component of the boundary of X carries a metricwith positive scalar curvature and (ii) either b_2^+(X)>0 or the boundary of Xis disconnected. As an application we show that the Poincare homology 3-sphere,oriented as the boundary of the positive E_8 plumbing, does not carrysymplectically semi-fillable contact structures. This proves, in particular, aconjecture of Gompf, and provides the first example of a 3-manifold which isnot symplectically semi-fillable. Using work of Froyshov, we also prove aresult constraining the topology of symplectic fillings of rational homology3-spheres having positive scalar curvature metrics.

Symplectic fillings and positive scalar curvature