摘要：

We present a structure theorem for cohomology groups of pseudo-effective line bundles over holomorphically convex Kähler manifolds, which generalizes the results of Takegoshi, Demailly-Peternell-Schneider, Meng-Zhou. As applications, we first give an answer to a question proposed by Matsumura, and establish an injectivity theorem for purely log terminal pairs generalized to pseudo-effective line bundles with transcendental singularities. Then we show a Kawamata-Viehweg-Kollár-Nadel type vanishing theorem for higher direct images in terms of numerical dimension for closed positive currents on compact Kähler manifolds. These are based on works with Prof. Xiankui Meng, Hongzhao Sun, and Prof. Xiangyu Zhou.