  We establish the cancellation of the first $2j$ terms in the diagonal
asymptotic expansion of the restriction to the $(0,2j)$-forms of the Bergman
kernel associated to the spin${}^c$ Dirac operator on high tensor powers of a
positive line bundle twisted by a (non necessarily holomorphic) complex vector
bundle, over a compact K\"{a}hler manifold. Moreover, we give a local formula
for the first and the second (non-zero) leading coefficients.
