the basic setup is a dimension 4 compact oriented Riemannian manifold, consider we have a closed 2-form $a$ on it with the cohomology class $[a]$.
Suppose now we have a restriction on the self-dual part of $a$, $$P_+a=b$$ Where $P_+$ is the projection of $a$ on self-dual part and $b$ is an arbitary self-dual two form.
How do we deduce from it the restriction of the cohomology class $$P_+(h[a])=h(b)$$ Where $h$ denotes the projection onto harmonic 2 forms, according to Hodge theory, this is well-defined on the cohomology class.
My attempt: using the fact that $P_+a=\frac{a+*a}{2}$, this problem can then be reduced to $h(*a)=*h(a)$, but I don't see how to prove this.