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