  We determine linear inequalities on the eigenvalues of curvature operators
that imply vanishing of the twisted $\hat A$ genus on a closed Riemannian spin
manifold, where the twisting bundle is any prescribed parallel bundle of
tensors. These inequalities yield surgery-stable curvature conditions tailored
to annihilate further rational cobordism invariants, such as the Witten genus,
elliptic genus, signature, and even the rational cobordism class itself.
