Exact anti-self-dual form on a compact four-manifold
= Exact anti-self-dual form on a compact four-manifold
If an exact two-form $\beta=d\alpha$ on a compact oriented Riemannian four-manifold is anti-self-dual, then
$$
0=\int_Md(\alpha\wedge d\alpha)=\int_M\beta\wedge\beta=-\lVert\beta\rVert_{L^2}^2,
$$
so $\beta=0$.