  We derive various pinching results for small Dirac eigenvalues using the
classification of $\text{spin}^c$ and spin manifolds admitting nontrivial
Killing spinors. For this, we introduce a notion of convergence for
$\text{spin}^c$ manifolds which involves a general study on convergence of
Riemannian manifolds with a principal $\mathbb{S}^1$-bundle. We also analyze
the relation between the regularity of the Riemannian metric and the regularity
of the curvature of the associated principal $\mathbb{S}^1$-bundle on
$\text{spin}^c$ manifolds with Killing spinors.
