  In this article, we generalize the classical Bochner-Weitzenb\"ock theorem
for manifolds satisfying an integral pinching on the curvature. We obtain the
vanishing of Betti numbers under integral pinching assumptions on the
curvature, and characterize the equality case. In particular, we reprove and
extend to higher degrees and higher dimensions a number of integral pinching
results obtained by M. Gursky for four-dimensional closed manifolds.
