Einstein metrics, harmonic forms, and symplectic four-manifolds

Abstract

If M is the underlying smooth oriented four-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics h on M such that (Formula Presented.) is the self-dual Weyl curvature of h, and ω is a non-trivial self-dual harmonic two-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.