  Let $M$ be an arbitrary complex manifold and let $L$ be a Hermitian
holomorphic line bundle over $M$. We introduce the Berezin-Toeplitz
quantization of the open set of $M$ where the curvature on $L$ is
non-degenerate. The quantum spaces are the spectral spaces corresponding to
$[0,k^{-N}]$ ($N>1$ fixed), of the Kodaira Laplace operator acting on forms
with values in tensor powers $L^k$. We establish the asymptotic expansion of
associated Toeplitz operators and their composition as $k\to\infty$ and we
define the corresponding star-product. If the Kodaira Laplace operator has a
certain spectral gap this method yields quantization by means of harmonic
forms. As applications, we obtain the Berezin-Toeplitz quantization for
semi-positive and big line bundles.
