Einstein Metrics, Harmonic Forms, and Symplectic Four-Manifolds

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 $M$.

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