A pr 2 01 5 Einstein Metrics , Harmonic Forms , and Symplectic Four - Manifolds Claude LeBrun

If M is the underlying smooth oriented 4-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics h on M such that W(ω, ω) > 0, where W is the self-dual Weyl curvature of h, and ω is a non-trivial self-dual harmonic 2-form on (M,h). While this open region in the space of Riemannian metrics contains all the known Einstein metrics on M , we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on M .