Embedding theorems for quantizable pseudo-Kähler manifolds

Given a compact quantizable pseudo-Kähler manifold $(M,\omega )$ of constant signature, there exists a Hermitian line bundle $(L,h)$ over $M$ with curvature $-2\pi i\omega$. We shall show that the asymptotic expansion of the Bergman kernels for $L^{\otimes k}$-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.