Bigness of adjoint linear subsystem and approximation theorems with ideal sheaves on weakly pseudoconvex manifolds

Abstract

Let $X$ be a weakly pseudoconvex manifold and $L\longrightarrow X$ be a holomorphic line bundle with a singular positive Hermitian metric $h$. In this article, we provide a points separation theorem and an embedding for the adjoint linear subsystem including the multiplier ideal sheaf $\mathscr{I}(h^m)$, with respect to an appropriate set excluding a singular locus of $h$. We also show that the adjoint bundle of $L$ is big, which constitutes a generalization to weakly pseudoconvex manifolds of Demailly's characterization of positivity in complex and algebraic geometry. To handle analytical methods, an approximation of singular Hermitian metrics is first constructed based on Demailly's approximation, using the strong openness property, preserving the ideal sheaves and compatible with blow-ups. Using the blow-ups obtained from this approximation, the singular holomorphic Morse inequalities and the approximation theorem for holomorphic sections, each twisted by the ideal sheaves, are established. This approximation theorem for sections provides the key to globalization, leading to global embeddings.