  In this paper, we study closed four-dimensional manifolds. In particular, we
show that under various new pinching curvature conditions (for example, the
sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then
the manifold is definite. If restricting to a metric with harmonic Weyl tensor,
then it must be self-dual or anti-self-dual under the same conditions.
Similarly, if restricting to an Einstein metric, then it must be either the
complex projective space with its Fubini-Study metric, the round sphere or
their quotients. Furthermore, we also classify Einstein manifolds with positive
intersection form and an upper bound on the sectional curvature.
