  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 $W^+(\omega ,
\omega )> 0$, where $W^+$ is the self-dual Weyl curvature of $h$, and $\omega$
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 $M$.
