  We obtain a vanishing theorem for the kernel of a Dirac operator on a
Clifford module twisted by a sufficiently large power of a line bundle, whose
curvature is non-degenerate at any point of the base manifold. In particular,
if the base manifold is almost complex, we prove a vanishing theorem for the
kernel of a $\spin^c$ Dirac operator twisted by a line bundle with curvature of
a mixed sign. In this case we also relax the assumption of non-degeneracy of
the curvature. These results are generalization of a vanishing theorem of
Borthwick and Uribe. As an application we obtain a new proof of the classical
Andreotti-Grauert vanishing theorem for the cohomology of a compact complex
manifold with values in the sheaf of holomorphic sections of a holomorphic
vector bundle, twisted by a large power of a holomorphic line bundle with
curvature of a mixed sign.
  As another application we calculate the sign of the index of a signature
operator twisted by a large power of a line bundle.
