 Exact anti-self-dual form on a compact four-manifold

Let $M$ be a compact oriented Riemannian manifold of dimension four, without boundary. An exact two-form $\beta =d\alpha$ that is an anti-self-dual differential form must vanish. Indeed the Generalized Stokes theorem and the Hodge star operator identity $\ast \beta =-\beta$ giveThe boundaryless hypothesis is essential. On a Euclidean four-ball, the nonzero constant two-form $d{x}^1\wedge d{x}^2-d{x}^3\wedge d{x}^4$ is both an exact differential form and an anti-self-dual differential form; the omitted boundary integral need not vanish there.