We derive various pinching results for small Dirac eigenvalues using the classification of spin^c and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for spin^c manifolds which involves a general study on convergence of Riemannian manifolds with a principal S¹-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal S¹-bundle on spin^c manifolds with Killing spinors.