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The first terms in the expansion of the Bergman kernel in higher degrees
We establish the cancellation of the first terms in the diagonal
asymptotic expansion of the restriction to the -forms of the Bergman
kernel associated to the spin 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
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