Embedding theorems for quantizable pseudo-Kähler manifolds

Given a compact quantizable pseudo-Kähler manifold \((M,\omega )\) $(M,\omega )$ of constant signature, there exists a Hermitian line bundle (L, h) over M with curvature \(-2\pi i\,\omega \) $-2\pi i\omega$ . We shall show that the asymptotic expansion of the Bergman kernels for \(L^{\otimes k}\) $L^{\otimes k}$ -valued (0, q)-forms implies more or less immediately a number of analogues of well-known results, such as Kodaira embedding theorem and Tian’s almost-isometry theorem.